Value Name Graphics Symbol LaTeX Formula Nº OEIS Continued fraction Year Web format 0.74048 04896 93061 04116 Hermite constant Sphere packing 3D Kepler conjecture μ K {\displaystyle {\mu _{_{K}}}} π 3 2 . . . . . . {\displaystyle {\frac {\pi }{3{\sqrt {2}}}}{\color {white}......\color {black}}} The Flyspeck project, led by Thomas Hales, demonstrated in 2014 that Kepler's conjecture is true. ਫਰਮਾ:OEIS2C [0;1,2,1,5,1,4,2,2,1,1,2,2,2,6,1,1,1,5,2,1,1,1, ...] 1611 0.74048048969306104116931349834344894 22.45915 77183 61045 47342 pi^e π e {\displaystyle \pi ^{e}} π e {\displaystyle \pi ^{e}} ਫਰਮਾ:OEIS2C [22;2,5,1,1,1,1,1,3,2,1,1,3,9,15,25,1,1,5,...] 22.4591577183610454734271522045437350 2.80777 02420 28519 36522 Fransén-Robinson constant F {\displaystyle {F}} ∫ 0 ∞ 1 Γ ( x ) d x . = e + ∫ 0 ∞ e − x π 2 + ln 2 x d x {\displaystyle \int _{0}^{\infty }{\frac {1}{\Gamma (x)}}\,dx.=e+\int _{0}^{\infty }{\frac {e^{-x}}{\pi ^{2}+\ln ^{2}x}}\,dx} N [ int [ 0 to ∞ ] { 1 / Gamma ( x )}] ਫਰਮਾ:OEIS2C [2;1,4,4,1,18,5,1,3,4,1,5,3,6,1,1,1,5,1,1,1...] 1978 2.80777024202851936522150118655777293 1.30568 6729 ≈ by Thomas & Dhar 1.30568 8 ≈ by McMullen Fractal dimension of the Apollonian packing of circles · ε {\displaystyle \varepsilon } ਫਰਮਾ:OEIS2C [0;3,2,3,16,8,10,3,1,1,2,1,3,1,2,13,1,1,4,1,5,...] 1994 1998 1.305686729 ≈ 1.305688 ≈ 0.43828 29367 27032 11162 0.36059 24718 71385 485 i
Infinite Tetration of i ∞ i {\displaystyle {}^{\infty }{i}} lim n → ∞ n i = lim n → ∞ i i ⋅ ⋅ i ⏟ n {\displaystyle \lim _{n\to \infty }{}^{n}i=\lim _{n\to \infty }\underbrace {i^{i^{\cdot ^{\cdot ^{i}}}}} _{n}} C ਫਰਮਾ:OEIS2C ਫਰਮਾ:OEIS2C [0;2,3,1,1,4,2,2,1,10,2,1,3,1,8,2,1,2,1, ...] + [0;2,1,3,2,2,3,1,5,5,1,2,1,10,10,6,1,1...] i 0.43828293672703211162697516355126482 + 0.36059247187138548595294052690600 i 0.92883 58271 Sum of the reciprocals of the averages of the twin prime pairs, JJGJJG B 1 {\displaystyle B_{1}} 1 4 + 1 6 + 1 12 + 1 18 + 1 30 + 1 42 + 1 60 + 1 72 + ⋯ {\displaystyle {\frac {1}{4}}+{\frac {1}{6}}+{\frac {1}{12}}+{\frac {1}{18}}+{\frac {1}{30}}+{\frac {1}{42}}+{\frac {1}{60}}+{\frac {1}{72}}+\cdots } 1 / 4 + 1 / 6 + 1 / 12 + 1 / 18 + 1 / 30 + 1 / 42 + 1 / 60 + 1 / 72 + ... ਫਰਮਾ:OEIS2C [0; 1, 13, 19, 4, 2, 3, 1, 1] 2014 0.928835827131 0.63092 97535 71457 43709 Fractal dimension of the Cantor set d f ( k ) {\displaystyle d_{f}(k)} lim ε → 0 log N ( ε ) log ( 1 / ε ) = log 2 log 3 {\displaystyle \lim _{\varepsilon \to 0}{\frac {\log N(\varepsilon )}{\log(1/\varepsilon )}}={\frac {\log 2}{\log 3}}} T ਫਰਮਾ:OEIS2C [0;1,1,1,2,2,3,1,5,2,23,2,2,1,1,55,1,4,3,1,1,...] 0.63092975357145743709952711434276085 0.31830 98861 83790 67153 ।nverse of Pi, Ramanujan 1 π {\displaystyle {\frac {1}{\pi }}} 2 2 9801 ∑ n = 0 ∞ ( 4 n ) ! ( 1103 + 26390 n ) ( n ! ) 4 396 4 n {\displaystyle {\frac {2{\sqrt {2}}}{9801}}\sum _{n=0}^{\infty }{\frac {(4n)!\,(1103+26390\;n)}{(n!)^{4}\,396^{4n}}}} 2 sqrt ( 2 ) / 9801 * Sum [ n = 0 to ∞ ] {(( 4 n ) !/ n !^ 4 ) * ( 1103 + 26390 n ) / 396 ^ ( 4 n )} T ਫਰਮਾ:OEIS2C [0;3,7,15,292,1,1,1,2,1,3,1,14,2,1,1,2,2,2,...] 0.31830988618379067153776752674502872 0.28878 80950 86602 42127 Flajolet and Richmond Q {\displaystyle {Q}} ∏ n = 1 ∞ ( 1 − 1 2 n ) = ( 1 − 1 2 1 ) ( 1 − 1 2 2 ) ( 1 − 1 2 3 ) . . . {\displaystyle \prod _{n=1}^{\infty }\left(1-{\frac {1}{2^{n}}}\right)=\left(1{-}{\frac {1}{2^{1}}}\right)\left(1{-}{\frac {1}{2^{2}}}\right)\left(1{-}{\frac {1}{2^{3}}}\right)...} ਫਰਮਾ:OEIS2C [0;3,2,6,4,1,2,1,9,2,1,2,3,2,3,5,1,2,1,1,6,1,...] 1992 0.28878809508660242127889972192923078 1.53960 07178 39002 03869 Lieb's square ice constant W 2 D {\displaystyle {W}_{2D}} lim n → ∞ ( f ( n ) ) n − 2 = ( 4 3 ) 3 2 = 8 3 3 {\displaystyle \lim _{n\to \infty }(f(n))^{n^{-2}}=\left({\frac {4}{3}}\right)^{\frac {3}{2}}={\frac {8}{3{\sqrt {3}}}}} A ਫਰਮਾ:OEIS2C [1;1,1,5,1,4,2,1,6,1,6,1,2,4,1,5,1,1,2,...] 1967 1.53960071783900203869106341467188655 0.20787 95763 50761 90854 i i {\displaystyle i^{i}} i i {\displaystyle i^{i}} e − π 2 {\displaystyle e^{-{\frac {\pi }{2}}}} T ਫਰਮਾ:OEIS2C [0;4,1,4,3,1,1,1,1,1,1,1,1,7,1,20,1,3,6,10,...] 1746 0.20787957635076190854695561983497877 4.53236 01418 27193 80962 Van der Pauw constant α {\displaystyle {\alpha }} π ln ( 2 ) = ∑ n = 0 ∞ 4 ( − 1 ) n 2 n + 1 ∑ n = 1 ∞ ( − 1 ) n + 1 n = 4 1 − 4 3 + 4 5 − 4 7 + 4 9 − ⋯ 1 1 − 1 2 + 1 3 − 1 4 + 1 5 − ⋯ {\displaystyle {\frac {\pi }{\ln(2)}}={\frac {\sum \limits _{n=0}^{\infty }{\frac {4(-1)^{n}}{2n+1}}}{\sum \limits _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n}}}}={\frac {{\frac {4}{1}}-{\frac {4}{3}}+{\frac {4}{5}}-{\frac {4}{7}}+{\frac {4}{9}}-\cdots }{{\frac {1}{1}}-{\frac {1}{2}}+{\frac {1}{3}}-{\frac {1}{4}}+{\frac {1}{5}}-\cdots }}} ਫਰਮਾ:OEIS2C [4;1,1,7,4,2,3,3,1,4,1,1,4,7,2,3,3,12,2,1,...] 4.53236014182719380962768294571666681 0.76159 41559 55764 88811 Hyperbolic tangent of 1 t h 1 {\displaystyle {th}\,1} − i tan ( i ) = e − 1 e e + 1 e = e 2 − 1 e 2 + 1 {\displaystyle -i\tan(i)={\frac {e-{\frac {1}{e}}}{e+{\frac {1}{e}}}}={\frac {e^{2}-1}{e^{2}+1}}} T ਫਰਮਾ:OEIS2C [0;1,3,5,7,9,11,13,15,17,19,21,23,25,27,...] = [0;2p+1 ], p∈ℕ 0.76159415595576488811945828260479359 0.59017 02995 08048 11302 Chebyshev constant · λ C h {\displaystyle {\lambda _{Ch}}} Γ ( 1 4 ) 2 4 π 3 / 2 = 4 ( 1 4 ! ) 2 π 3 / 2 {\displaystyle {\frac {\Gamma ({\tfrac {1}{4}})^{2}}{4\pi ^{3/2}}}={\frac {4({\tfrac {1}{4}}!)^{2}}{\pi ^{3/2}}}} ( Gamma ( 1 / 4 ) ^ 2 ) / ( 4 pi ^ ( 3 / 2 )) ਫਰਮਾ:OEIS2C [0;1,1,2,3,1,2,41,1,6,5,124,5,2,2,1,1,6,1,2,...] 0.59017029950804811302266897027924429 0.07077 60393 11528 80353 -0.68400 03894 37932 129 i
MKB constant · · M I {\displaystyle M_{I}} lim n → ∞ ∫ 1 2 n ( − 1 ) x x x d x = ∫ 1 2 n e i π x x 1 / x d x {\displaystyle \lim _{n\rightarrow \infty }\int _{1}^{2n}(-1)^{x}~{\sqrt[{x}]{x}}~dx=\int _{1}^{2n}e^{i\pi x}~x^{1/x}~dx} lim_ ( 2 n -> ∞ ) int [ 1 to 2 n ] { exp ( i * Pi * x ) * x ^ ( 1 / x ) dx } C ਫਰਮਾ:OEIS2C ਫਰਮਾ:OEIS2C [0;14,7,1,2,1,23,2,1,8,16,1,1,3,1,26,1,6,1,1, ...] - [0;1,2,6,13,41,112,1,25,1,1,1,1,3,13,2,1, ...] i 2009 0.07077603931152880353952802183028200 -0.68400038943793212918274445999266 i 1.25992 10498 94873 16476 Cube root of 2 Delian Constant 2 3 {\displaystyle {\sqrt[{3}]{2}}} 2 3 {\displaystyle {\sqrt[{3}]{2}}} A ਫਰਮਾ:OEIS2C [1;3,1,5,1,1,4,1,1,8,1,14,1,10,2,1,4,12,2,3,...] 1.25992104989487316476721060727822835 1.09317 04591 95490 89396 Smarandache Constant 1ª S 1 {\displaystyle {S_{1}}} ∑ n = 2 ∞ 1 μ ( n ) ! . . . . {\displaystyle \sum _{n=2}^{\infty }{\frac {1}{\mu (n)!}}{\color {white}....\color {black}}} where μ (n ) is the Kempner function ਫਰਮਾ:OEIS2C [1;10,1,2,1,2,1,13,3,1,6,1,2,11,4,6,2,15,1,1,...] 1.09317045919549089396820137014520832 0.62481 05338 43826 58687 + 1.30024 25902 20120 419 i Generalized continued fraction of i F C G ( i ) {\displaystyle {{F}_{CG}}_{(i)}} i + i i + i i + i i + i i + i i + i i + i / . . . = 17 − 1 8 + i ( 1 2 + 2 17 − 1 ) {\displaystyle \textstyle i{+}{\frac {i}{i+{\frac {i}{i+{\frac {i}{i+{\frac {i}{i+{\frac {i}{i+{\frac {i}{i+i{/...}}}}}}}}}}}}}={\sqrt {\frac {{\sqrt {17}}-1}{8}}}+i\left({\tfrac {1}{2}}{+}{\sqrt {\frac {2}{{\sqrt {17}}-1}}}\right)} i + i / ( i + i / ( i + i / ( i + i / ( i + i / ( i + i / ( i + i / ( i + i / ( i + i / ( i + i / ( i + i / ( i + i / ( i + i / ( i + i / ( i + i / ( i + i / ( i + i / ( i + i / ( i + i / ( i + i / ( ... ))))))))))))))))))))) C A ਫਰਮਾ:OEIS2C ਫਰਮਾ:OEIS2C [i;1,i,1,i,1,i,1,i,1,i,1,i,1,i,1,i,1,i,1,i,1,i,1,i,1,i,1,i,1,..] = [0;1,i ] 0.62481053384382658687960444744285144 + 1.30024259022012041915890982074952 i 3.05940 74053 42576 14453 Double factorial constant C n ! ! {\displaystyle {C_{_{n!!}}}} ∑ n = 0 ∞ 1 n ! ! = e [ 1 2 + γ ( 1 2 , 1 2 ) ] {\displaystyle \sum _{n=0}^{\infty }{\frac {1}{n!!}}={\sqrt {e}}\left[{\frac {1}{\sqrt {2}}}+\gamma ({\tfrac {1}{2}},{\tfrac {1}{2}})\right]} ਫਰਮਾ:OEIS2C [3;16,1,4,1,66,10,1,1,1,1,2,5,1,2,1,1,1,1,1,2,...] 3.05940740534257614453947549923327861 5.97798 68121 78349 12266 Madelung Constant 2 H 2 ( 2 ) {\displaystyle {H}_{2}(2)} π ln ( 3 ) 3 {\displaystyle \pi \ln(3){\sqrt {3}}} ਫਰਮਾ:OEIS2C [5;1,44,2,2,1,15,1,1,12,1,65,11,1,3,1,1,...] 5.97798681217834912266905331933922774 0.91893 85332 04672 74178 Raabe's formula ζ ′ ( 0 ) {\displaystyle {\zeta '(0)}} ∫ a a + 1 log Γ ( t ) d t = 1 2 log 2 π + a log a − a , a ≥ 0 {\displaystyle \int \limits _{a}^{a+1}\log \Gamma (t)\,\mathrm {d} t={\tfrac {1}{2}}\log 2\pi +a\log a-a,\quad a\geq 0} integral_a ^ ( a + 1 ) { log ( Gamma ( x )) + a - a log ( a )} dx ਫਰਮਾ:OEIS2C [0;1,11,2,1,36,1,1,3,3,5,3,1,18,2,1,1,2,2,1,1,...] 0.91893853320467274178032973640561763 2.20741 60991 62477 96230 Lower limit in the moving sofa problem S H {\displaystyle {S_{_{H}}}} π 2 + 2 π {\displaystyle {\frac {\pi }{2}}+{\frac {2}{\pi }}} T ਫਰਮਾ:OEIS2C [2;4,1,4,1,1,2,5,1,11,1,1,5,1,6,1,3,1,1,1,1,7,...] 1967 2.20741609916247796230685674512980889 1.17628 08182 59917 50654 Salem number, Lehmer's conjecture
σ 10 {\displaystyle {\sigma _{_{10}}}} x 10 + x 9 − x 7 − x 6 − x 5 − x 4 − x 3 + x + 1 {\displaystyle x^{10}+x^{9}-x^{7}-x^{6}-x^{5}-x^{4}-x^{3}+x+1} x ^ 10 + x ^ 9 - x ^ 7 - x ^ 6 - x ^ 5 - x ^ 4 - x ^ 3 + x + 1 A ਫਰਮਾ:OEIS2C [1;5,1,2,17,1,7,2,1,1,2,4,7,2,2,1,1,15,1,1, ... 1983? 1.17628081825991750654407033847403505 0.37395 58136 19202 28805 Artin constant C A r t i n {\displaystyle {C}_{Artin}} ∏ n = 1 ∞ ( 1 − 1 p n ( p n − 1 ) ) p n = prime {\displaystyle \prod _{n=1}^{\infty }\left(1-{\frac {1}{p_{n}(p_{n}-1)}}\right)\quad p_{n}\scriptstyle {\text{ = prime}}} Prod [ n = 1 to ∞ ] { 1-1 / ( prime ( n ) ( prime ( n ) -1 ))} ਫਰਮਾ:OEIS2C [0;2,1,2,14,1,1,2,3,5,1,3,1,5,1,1,2,3,5,46,...] 1999 0.37395581361920228805472805434641641 0.42215 77331 15826 62702 Volume of Reuleaux tetrahedron V R {\displaystyle {V_{_{R}}}} s 3 12 ( 3 2 − 49 π + 162 arctan 2 ) {\displaystyle {\frac {s^{3}}{12}}(3{\sqrt {2}}-49\,\pi +162\,\arctan {\sqrt {2}})} ( 3 * Sqrt [ 2 ] - 49 * Pi + 162 * ArcTan [ Sqrt [ 2 ]]) / 12 ਫਰਮਾ:OEIS2C [0;2,2,1,2,2,7,4,4,287,1,6,1,2,1,8,5,1,1,1,1, ...] 0.42215773311582662702336591662385075 2.82641 99970 67591 57554 Murata Constant C m {\displaystyle {C_{m}}} ∏ n = 1 ∞ ( 1 + 1 ( p n − 1 ) 2 ) p n : p r i m e {\displaystyle \prod _{n=1}^{\infty }{\underset {p_{n}:\,{prime}}{{\Big (}1+{\frac {1}{(p_{n}-1)^{2}}}{\Big )}}}} Prod [ n = 1 to ∞ ] { 1 + 1 / ( prime ( n ) -1 ) ^ 2 } ਫਰਮਾ:OEIS2C [2;1,4,1,3,5,2,2,2,4,3,2,1,3,2,1,1,1,8,2,2,28,...] 2.82641999706759157554639174723695374 1.09864 19643 94156 48573 Paris Constant C P a {\displaystyle C_{Pa}} ∏ n = 2 ∞ 2 φ φ + φ n , φ = F i {\displaystyle \prod _{n=2}^{\infty }{\frac {2\varphi }{\varphi +\varphi _{n}}}\;,\;\varphi {=}{Fi}} con φ n = 1 + φ n − 1 {\displaystyle \varphi _{n}{=}{\sqrt {1{+}\varphi _{n{-}1}}}} y φ 1 = 1 {\displaystyle \varphi _{1}{=}1} ਫਰਮਾ:OEIS2C [1;10,7,3,1,3,1,5,1,4,2,7,1,2,3,22,1,2,5,2,1,...] 1.09864196439415648573466891734359621 2.39996 32297 28653 32223 Radians Golden angle b {\displaystyle {b}} ( 4 − 2 Φ ) π = ( 3 − 5 ) π {\displaystyle (4-2\,\Phi )\,\pi =(3-{\sqrt {5}})\,\pi } = 137.5077640500378546 ...° T ਫਰਮਾ:OEIS2C [2;2,1,1,1087,4,4,120,2,1,1,2,1,1,7,7,2,11,...] 1907 2.39996322972865332223155550663361385 1.64218 84352 22121 13687 Lebesgue constant L2 L 2 {\displaystyle {L2}} 1 5 + 25 − 2 5 π = 1 π ∫ 0 π | sin ( 5 t 2 ) | sin ( t 2 ) d t {\displaystyle {\frac {1}{5}}+{\frac {\sqrt {25-2{\sqrt {5}}}}{\pi }}={\frac {1}{\pi }}\int _{0}^{\pi }{\frac {\left|\sin({\frac {5t}{2}})\right|}{\sin({\frac {t}{2}})}}\,dt} 1 / 5 + sqrt ( 25 - 2 * sqrt ( 5 )) / Pi T ਫਰਮਾ:OEIS2C [1;1,1,1,3,1,6,1,5,2,2,3,1,2,7,1,3,5,2,2,1,1,...] 1910 1.64218843522212113687362798892294034 1.26408 47353 05301 11307 Vardi constant V c {\displaystyle {V_{c}}} 3 2 ∏ n ≥ 1 ( 1 + 1 ( 2 e n − 1 ) 2 ) 1 / 2 n + 1 {\displaystyle {\frac {\sqrt {3}}{\sqrt {2}}}\prod _{n\geq 1}\left(1+{1 \over (2e_{n}-1)^{2}}\right)^{\!1/2^{n+1}}} ਫਰਮਾ:OEIS2C [1;3,1,3,1,2,5,54,7,1,2,1,2,3,15,1,2,1,1,2,1,...] 1991 1.26408473530530111307959958416466949 1.5065918849 ± 0.0000000028 Area of the Mandelbrot fractal γ {\displaystyle \gamma } This is conjectured to be: 6 π − 1 − e = 1.506591651 ⋯ {\displaystyle {\sqrt {6\pi -1}}-e=1.506591651\cdots } ਫਰਮਾ:OEIS2C [1;1,1,37,2,2,1,10,1,1,2,2,4,1,1,1,1,5,4,...] 1912 1.50659177 +/- 0.00000008 1.61111 49258 08376 736 111···111 27224 36828 183213 ones Exponential factorial constant S E f {\displaystyle {S_{Ef}}} ∑ n = 1 ∞ 1 n ( n − 1 ) ⋅ ⋅ ⋅ 2 1 = 1 + 1 2 1 + 1 3 2 1 + 1 4 3 2 1 + 1 5 4 3 2 1 + ⋯ {\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n^{(n{-}1)^{\cdot ^{\cdot ^{\cdot ^{2^{1}}}}}}}}=1{+}{\frac {1}{2^{1}}}{+}{\frac {1}{3^{2^{1}}}}+{\frac {1}{4^{3^{2^{1}}}}}+{\frac {1}{5^{4^{3^{2^{1}}}}}}{+}\cdots } T ਫਰਮਾ:OEIS2C [1; 1, 1, 1, 1, 2, 1, 808, 2, 1, 2, 1, 14,...] 1.61111492580837673611111111111111111 1.11786 41511 89944 97314 Goh-Schmutz constant C G S {\displaystyle C_{GS}} ∫ 0 ∞ log ( s + 1 ) e s − 1 d s = − ∑ n = 1 ∞ e n n E i ( − n ) {\displaystyle \int _{0}^{\infty }{\frac {\log(s+1)}{e^{s}-1}}\ ds=\!-\!\sum _{n=1}^{\infty }{\frac {e^{n}}{n}}Ei(-n)} Integrate { log ( s + 1 ) / ( E ^ s -1 )} ਫਰਮਾ:OEIS2C [1;8,2,15,2,7,2,1,1,1,1,2,3,5,3,5,1,1,4,13,1,...] 1.11786415118994497314040996202656544 0.31813 15052 04764 13531 ±1.33723 57014 30689 40 i
Fixed points Super-Logarithm · Tetration − W ( − 1 ) {\displaystyle {-W(-1)}} lim n → ∞ {\displaystyle \lim _{n\rightarrow \infty }} f ( x ) = log ( log ( log ( log ( ⋯ log ( log ( x ) ) ) ) ) ) ⏟ log s n times {\displaystyle f(x)=\underbrace {\log(\log(\log(\log(\cdots \log(\log(x))))))\,\!} \atop {\log _{s}{\text{ n times}}}} For an initial value of x different to 0 , 1 , e , e e , e e e {\textstyle 0,1,e,e^{e},e^{e^{e}}} , etc.
-W(-1)
where W=ProductLog Lambert W function
C ਫਰਮਾ:OEIS2C ਫਰਮਾ:OEIS2C [-i;1 +2i,1+i,6-i,1+2i,-7+3i,2i,2,1-2i,-1+i,-, ...] 0.31813150520476413531265425158766451 -1.33723570143068940890116214319371 i 0.28016 94990 23869 13303 Bernstein's constant β {\displaystyle {\beta }} ≈ 1 2 π {\displaystyle \approx {\frac {1}{2{\sqrt {\pi }}}}} T ਫਰਮਾ:OEIS2C [0;3,1,1,3,9,6,3,1,3,14,34,2,1,1,60,2,2,1,1,...] 1913 0.28016949902386913303643649123067200 0.66016 18158 46869 57392 Twin Primes Constant C 2 {\displaystyle {C}_{2}} ∏ p = 3 ∞ p ( p − 2 ) ( p − 1 ) 2 {\displaystyle \prod _{p=3}^{\infty }{\frac {p(p-2)}{(p-1)^{2}}}} prod [ p = 3 to ∞ ] { p ( p -2 ) / ( p -1 ) ^ 2 ਫਰਮਾ:OEIS2C [0;1,1,1,16,2,2,2,2,1,18,2,2,11,1,1,2,4,1,...] 1922 0.66016181584686957392781211001455577 1.22674 20107 20353 24441 Fibonacci Factorial constant F {\displaystyle F} ∏ n = 1 ∞ ( 1 − ( − 1 φ 2 ) n ) = ∏ n = 1 ∞ ( 1 − ( 5 − 3 2 ) n ) {\displaystyle \prod _{n=1}^{\infty }\left(1-\left(-{\frac {1}{{\varphi }^{2}}}\right)^{n}\right)=\prod _{n=1}^{\infty }\left(1-\left({\frac {{\sqrt {5}}-3}{2}}\right)^{n}\right)} prod [ n = 1 to ∞ ] { 1 - (( sqrt ( 5 ) -3 ) / 2 ) ^ n } ਫਰਮਾ:OEIS2C [1;4,2,2,3,2,15,9,1,2,1,2,15,7,6,21,3,5,1,23,...] 1.22674201072035324441763023045536165 0.11494 20448 53296 20070 Kepler–Bouwkamp constant ρ {\displaystyle {\rho }} ∏ n = 3 ∞ cos ( π n ) = cos ( π 3 ) cos ( π 4 ) cos ( π 5 ) . . . {\displaystyle \prod _{n=3}^{\infty }\cos \left({\frac {\pi }{n}}\right)=\cos \left({\frac {\pi }{3}}\right)\cos \left({\frac {\pi }{4}}\right)\cos \left({\frac {\pi }{5}}\right)...} prod [ n = 3 to ∞ ] { cos ( pi / n )} ਫਰਮਾ:OEIS2C [0;8,1,2,2,1,272,2,1,41,6,1,3,1,1,26,4,1,1,...] 0.11494204485329620070104015746959874 1.78723 16501 82965 93301 Komornik–Loreti constant q {\displaystyle {q}} 1 = ∑ n = 1 ∞ t k q k Raiz real de ∏ n = 0 ∞ ( 1 − 1 q 2 n ) + q − 2 q − 1 = 0 {\displaystyle 1=\!\sum _{n=1}^{\infty }{\frac {t_{k}}{q^{k}}}\qquad \scriptstyle {\text{Raiz real de}}\displaystyle \prod _{n=0}^{\infty }\!\left(\!1{-}{\frac {1}{q^{2^{n}}}}\!\right)\!{+}{\frac {q{-}2}{q{-}1}}=0} tk = Thue–Morse sequence
FindRoot [( prod [ n = 0 to ∞ ] { 1-1 / ( x ^ 2 ^ n )} + ( x -2 ) / ( x -1 )) = 0 , { x , 1.7 }, WorkingPrecision -> 30 ] T ਫਰਮਾ:OEIS2C [1;1,3,1,2,3,188,1,12,1,1,22,33,1,10,1,1,7,...] 1998 1.78723165018296593301327489033700839 3.30277 56377 31994 64655 Bronze ratio σ R r {\displaystyle {\sigma }_{\,Rr}} 3 + 13 2 = 1 + 3 + 3 + 3 + 3 + ⋯ {\displaystyle {\frac {3+{\sqrt {13}}}{2}}=1+{\sqrt {3+{\sqrt {3+{\sqrt {3+{\sqrt {3+\cdots }}}}}}}}} A ਫਰਮਾ:OEIS2C [3;3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,...] = [3;3 ,...] 3.30277563773199464655961063373524797 0.82699 33431 32688 07426 Disk Covering C 5 {\displaystyle {C_{5}}} 1 ∑ n = 0 ∞ 1 ( 3 n + 2 2 ) = 3 3 2 π {\displaystyle {\frac {1}{\sum \limits _{n=0}^{\infty }{\frac {1}{\binom {3n+2}{2}}}}}={\frac {3{\sqrt {3}}}{2\pi }}} T ਫਰਮਾ:OEIS2C [0;1,4,1,3,1,1,4,1,2,2,1,1,7,1,4,4,2,1,1,1,1,...] 1939 1949 0.82699334313268807426698974746945416 2.66514 41426 90225 18865 Gelfond–Schneider constant G G S {\displaystyle G_{\,GS}} 2 2 {\displaystyle 2^{\sqrt {2}}} T ਫਰਮਾ:OEIS2C [2;1,1,1,72,3,4,1,3,2,1,1,1,14,1,2,1,1,3,1,...] 1934 2.66514414269022518865029724987313985 3.27582 29187 21811 15978 Khinchin-Lévy constant γ {\displaystyle \gamma } e π 2 / ( 12 ln 2 ) {\displaystyle e^{\pi ^{2}/(12\ln 2)}} ਫਰਮਾ:OEIS2C [3;3,1,1,1,2,29,1,130,1,12,3,8,2,4,1,3,55,...] 1936 3.27582291872181115978768188245384386 0.52382 25713 89864 40645 Chi Function Hyperbolic cosine integral C h i ( ) {\displaystyle {\operatorname {Chi()} }} γ + ∫ 0 x cosh t − 1 t d t {\displaystyle \gamma +\int _{0}^{x}{\frac {\cosh t-1}{t}}\,dt} γ = Euler–Mascheroni constant= 0.5772156649... {\displaystyle \scriptstyle \gamma \,{\text{= Euler–Mascheroni constant= 0.5772156649...}}}
ਫਰਮਾ:OEIS2C [0;1,1,9,1,172,1,7,1,11,1,1,2,1,8,1,1,1,1,1,...] 0.52382257138986440645095829438325566 1.13198 82487 943 Viswanath constant C V i {\displaystyle {C}_{Vi}} lim n → ∞ | a n | 1 n {\displaystyle \lim _{n\to \infty }|a_{n}|^{\frac {1}{n}}} where an = Fibonacci sequence T ? ਫਰਮਾ:OEIS2C [1;7,1,1,2,1,3,2,1,2,1,8,1,5,1,1,1,9,1,...] 1997 1.1319882487943 ... 1.23370 05501 36169 82735 Favard constant 3 4 ζ ( 2 ) {\displaystyle {\tfrac {3}{4}}\zeta (2)} π 2 8 = ∑ n = 0 ∞ 1 ( 2 n − 1 ) 2 = 1 1 2 + 1 3 2 + 1 5 2 + 1 7 2 + ⋯ {\displaystyle {\frac {\pi ^{2}}{8}}=\sum _{n=0}^{\infty }{\frac {1}{(2n-1)^{2}}}={\frac {1}{1^{2}}}+{\frac {1}{3^{2}}}+{\frac {1}{5^{2}}}+{\frac {1}{7^{2}}}+\cdots } sum [ n = 1 to ∞ ] { 1 / (( 2 n -1 ) ^ 2 )} T ਫਰਮਾ:OEIS2C [1;4,3,1,1,2,2,5,1,1,1,1,2,1,2,1,10,4,3,1,1,...] 1902 a 1965 1.23370055013616982735431137498451889 2.50662 82746 31000 50241 Square root of 2 pi 2 π {\displaystyle {\sqrt {2\pi }}} 2 π = lim n → ∞ n ! e n n n n . . . . {\displaystyle {\sqrt {2\pi }}=\lim _{n\to \infty }{\frac {n!\;e^{n}}{n^{n}{\sqrt {n}}}}{\color {white}....\color {black}}} Stirling's approximation T ਫਰਮਾ:OEIS2C [2;1,1,37,4,1,1,1,1,9,1,1,2,8,6,1,2,2,1,3,...] 1692 a 1770 2.50662827463100050241576528481104525 4.13273 13541 22492 93846 Square root of Tau·e τ e {\displaystyle {\sqrt {\tau e}}} 2 π e {\displaystyle {\sqrt {2\pi e}}} ਫਰਮਾ:OEIS2C [4;7,1,1,6,1,5,1,1,1,8,3,1,2,2,15,2,1,1,2,4,...] 4.13273135412249293846939188429985264 0.97027 01143 92033 92574 Lochs constant £ L o {\displaystyle {{\text{£}}_{_{Lo}}}} 6 ln 2 ln 10 π 2 {\displaystyle {\frac {6\ln 2\ln 10}{\pi ^{2}}}} ਫਰਮਾ:OEIS2C [0;1,32,1,1,1,2,1,46,7,2,7,10,8,1,71,1,37,1,1,...] 1964 0.97027011439203392574025601921001083 0.98770 03907 36053 46013 Area bounded by the eccentric rotation of Reuleaux triangle T R {\displaystyle {\mathcal {T}}_{R}} a 2 ⋅ ( 2 3 + π 6 − 3 ) {\displaystyle a^{2}\cdot \left(2{\sqrt {3}}+{\frac {\pi }{6}}-3\right)} where a = side length of the square T ਫਰਮਾ:OEIS2C [0;1,80,3,3,2,1,1,1,4,2,2,1,1,1,8,1,2,10,1,2,...] 1914 0.98770039073605346013199991355832854 0.70444 22009 99165 59273 Carefree constant 2 C 2 {\displaystyle {\mathcal {C}}_{2}} ∏ n = 1 ∞ ( 1 − 1 p n ( p n + 1 ) ) p n : p r i m e {\displaystyle {\underset {p_{n}:\,{prime}}{\prod _{n=1}^{\infty }\left(1-{\frac {1}{p_{n}(p_{n}+1)}}\right)}}} N [ prod [ n = 1 to ∞ ] { 1 - 1 / ( prime ( n ) * ( prime ( n ) + 1 ))}] ਫਰਮਾ:OEIS2C [0;1,2,2,1,1,1,1,4,2,1,1,3,703,2,1,1,1,3,5,1,...] 0.70444220099916559273660335032663721 1.84775 90650 22573 51225 Connective constant μ {\displaystyle {\mu }} 2 + 2 = lim n → ∞ c n 1 / n {\displaystyle {\sqrt {2+{\sqrt {2}}}}\;=\lim _{n\rightarrow \infty }c_{n}^{1/n}} as a root of the polynomial : x 4 − 4 x 2 + 2 = 0 {\displaystyle :\;x^{4}-4x^{2}+2=0}
A ਫਰਮਾ:OEIS2C [1;1,5,1,1,3,6,1,3,3,10,10,1,1,1,5,2,3,1,1,3,...] 1.84775906502257351225636637879357657 0.30366 30028 98732 65859 Gauss-Kuzmin-Wirsing constant λ 2 {\displaystyle {\lambda }_{2}} lim n → ∞ F n ( x ) − ln ( 1 − x ) ( − λ ) n = Ψ ( x ) , {\displaystyle \lim _{n\to \infty }{\frac {F_{n}(x)-\ln(1-x)}{(-\lambda )^{n}}}=\Psi (x),} where Ψ ( x ) {\displaystyle \Psi (x)} is an analytic function with Ψ ( 0 ) = Ψ ( 1 ) = 0 {\displaystyle \Psi (0)\!=\!\Psi (1)\!=\!0} .
ਫਰਮਾ:OEIS2C [0;3,3,2,2,3,13,1,174,1,1,1,2,2,2,1,1,1,2,2,1,...] 1973 0.30366300289873265859744812190155623 1.57079 63267 94896 61923 Favard constant K1 Wallis product π 2 {\displaystyle {\frac {\pi }{2}}} ∏ n = 1 ∞ ( 4 n 2 4 n 2 − 1 ) = 2 1 ⋅ 2 3 ⋅ 4 3 ⋅ 4 5 ⋅ 6 5 ⋅ 6 7 ⋅ 8 7 ⋅ 8 9 ⋯ {\displaystyle \prod _{n=1}^{\infty }\left({\frac {4n^{2}}{4n^{2}-1}}\right)={\frac {2}{1}}\cdot {\frac {2}{3}}\cdot {\frac {4}{3}}\cdot {\frac {4}{5}}\cdot {\frac {6}{5}}\cdot {\frac {6}{7}}\cdot {\frac {8}{7}}\cdot {\frac {8}{9}}\cdots } Prod [ n = 1 to ∞ ] {( 4 n ^ 2 ) / ( 4 n ^ 2-1 )} T ਫਰਮਾ:OEIS2C [1;1,1,3,31,1,145,1,4,2,8,1,6,1,2,3,1,4,1,5,1...] 1655 1.57079632679489661923132169163975144 1.60669 51524 15291 76378 Erdős–Borwein constant E B {\displaystyle {E}_{\,B}} ∑ m = 1 ∞ ∑ n = 1 ∞ 1 2 m n = ∑ n = 1 ∞ 1 2 n − 1 = 1 1 + 1 3 + 1 7 + 1 15 + . . . {\displaystyle \sum _{m=1}^{\infty }\sum _{n=1}^{\infty }{\frac {1}{2^{mn}}}=\sum _{n=1}^{\infty }{\frac {1}{2^{n}-1}}={\frac {1}{1}}\!+\!{\frac {1}{3}}\!+\!{\frac {1}{7}}\!+\!{\frac {1}{15}}\!+\!...} sum [ n = 1 to ∞ ] { 1 / ( 2 ^ n -1 )} I ਫਰਮਾ:OEIS2C [1;1,1,1,1,5,2,1,2,29,4,1,2,2,2,2,6,1,7,1,...] 1949 1.60669515241529176378330152319092458 1.61803 39887 49894 84820 Phi, Golden ratio φ {\displaystyle {\varphi }} 1 + 5 2 = 1 + 1 + 1 + 1 + ⋯ {\displaystyle {\frac {1+{\sqrt {5}}}{2}}={\sqrt {1+{\sqrt {1+{\sqrt {1+{\sqrt {1+\cdots }}}}}}}}} A ਫਰਮਾ:OEIS2C [0;1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,...] = [0;1 ,...] -300 ~ 1.61803398874989484820458683436563811 1.64493 40668 48226 43647 Riemann Function Zeta(2) ζ ( 2 ) {\displaystyle {\zeta }(\,2)} π 2 6 = ∑ n = 1 ∞ 1 n 2 = 1 1 2 + 1 2 2 + 1 3 2 + 1 4 2 + ⋯ {\displaystyle {\frac {\pi ^{2}}{6}}=\sum _{n=1}^{\infty }{\frac {1}{n^{2}}}={\frac {1}{1^{2}}}+{\frac {1}{2^{2}}}+{\frac {1}{3^{2}}}+{\frac {1}{4^{2}}}+\cdots } T ਫਰਮਾ:OEIS2C [1;1,1,1,4,2,4,7,1,4,2,3,4,10 1,2,1,1,1,15,...] 1826 to 1866 1.64493406684822643647241516664602519 1.73205 08075 68877 29352 Theodorus constant 3 {\displaystyle {\sqrt {3}}} 3 3 3 3 3 ⋯ 3 3 3 3 3 {\displaystyle {\sqrt[{3}]{3\,{\sqrt[{3}]{3\,{\sqrt[{3}]{3\,{\sqrt[{3}]{3\,{\sqrt[{3}]{3\,\cdots }}}}}}}}}}} ( 3 ( 3 ( 3 ( 3 ( 3 ( 3 ( 3 ) ^ 1 / 3 ) ^ 1 / 3 ) ^ 1 / 3 ) ^ 1 / 3 ) ^ 1 / 3 ) ^ 1 / 3 ) ^ 1 / 3 ... A ਫਰਮਾ:OEIS2C [1;1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,...] = [1;1,2 ,...] -465 to -398 1.73205080756887729352744634150587237 1.75793 27566 18004 53270 Kasner number R {\displaystyle {R}} 1 + 2 + 3 + 4 + ⋯ {\displaystyle {\sqrt {1+{\sqrt {2+{\sqrt {3+{\sqrt {4+\cdots }}}}}}}}} Fold [ Sqrt [ #1 + #2 ] & , 0 , Reverse [ Range [ 20 ]]] ਫਰਮਾ:OEIS2C [1;1,3,7,1,1,1,2,3,1,4,1,1,2,1,2,20,1,2,2,...] 1878 a 1955 1.75793275661800453270881963821813852 2.29558 71493 92638 07403 Universal parabolic constant P 2 {\displaystyle {P}_{\,2}} ln ( 1 + 2 ) + 2 = arcsinh ( 1 ) + 2 {\displaystyle \ln(1+{\sqrt {2}})+{\sqrt {2}}\;=\;\operatorname {arcsinh} (1)+{\sqrt {2}}} T ਫਰਮਾ:OEIS2C [2;3,2,1,1,1,1,3,3,1,1,4,2,3,2,7,1,6,1,8,7,2,1,...] 2.29558714939263807403429804918949038 1.78657 64593 65922 46345 Silverman constant S m {\displaystyle {{\mathcal {S}}_{_{m}}}} ∑ n = 1 ∞ 1 ϕ ( n ) σ 1 ( n ) = ∏ n = 1 ∞ ( 1 + ∑ k = 1 ∞ 1 p n 2 k − p n k − 1 ) p n : p r i m e {\displaystyle \sum _{n=1}^{\infty }{\frac {1}{\phi (n)\sigma _{1}(n)}}={\underset {p_{n}:\,{prime}}{\prod _{n=1}^{\infty }\left(1+\sum _{k=1}^{\infty }{\frac {1}{p_{n}^{2k}-p_{n}^{k-1}}}\right)}}} ø () = Euler's totient function, σ 1 () = Divisor function. Sum [ n = 1 to ∞ ] { 1 / [ EulerPhi ( n ) DivisorSigma ( 1 , n )]} ਫਰਮਾ:OEIS2C [1;1,3,1,2,5,1,65,11,2,1,2,13,1,4,1,1,1,2,5,4,...] 1.78657645936592246345859047554131575 2.59807 62113 53315 94029 Area of the regular hexagon with side equal to 1 A 6 {\displaystyle {\mathcal {A}}_{6}} 3 3 2 {\displaystyle {\frac {3{\sqrt {3}}}{2}}} A ਫਰਮਾ:OEIS2C [2;1,1,2,20,2,1,1,4,1,1,2,20,2,1,1,4,1,1,2,20,...] [2;1,1,2,20,2,1,1,4 ] 2.59807621135331594029116951225880855 0.66131 70494 69622 33528 Feller-Tornier constant C F T {\displaystyle {{\mathcal {C}}_{_{FT}}}} 1 2 ∏ n = 1 ∞ ( 1 − 2 p n 2 ) + 1 2 p n : p r i m e = 3 π 2 ∏ n = 1 ∞ ( 1 − 1 p n 2 − 1 ) + 1 2 {\displaystyle {\underset {p_{n}:\,{prime}}{{\frac {1}{2}}\prod _{n=1}^{\infty }\left(1-{\frac {2}{p_{n}^{2}}}\right){+}{\frac {1}{2}}}}={\frac {3}{\pi ^{2}}}\prod _{n=1}^{\infty }\left(1-{\frac {1}{p_{n}^{2}-1}}\right){+}{\frac {1}{2}}} [ prod [ n = 1 to ∞ ] { 1-2 / prime ( n ) ^ 2 }] / 2 + 1 / 2 T ? ਫਰਮਾ:OEIS2C [0;1,1,1,20,9,1,2,5,1,2,3,2,3,38,8,1,16,2,2,...] 1932 0.66131704946962233528976584627411853 1.46099 84862 06318 35815 Baxter's Four-coloring constant Mapamundi Four-Coloring C 2 {\displaystyle {\mathcal {C}}^{2}} ∏ n = 1 ∞ ( 3 n − 1 ) 2 ( 3 n − 2 ) ( 3 n ) = 3 4 π 2 Γ ( 1 3 ) 3 {\displaystyle \prod _{n=1}^{\infty }{\frac {(3n-1)^{2}}{(3n-2)(3n)}}={\frac {3}{4\pi ^{2}}}\,\Gamma \left({\frac {1}{3}}\right)^{3}} Γ() = Gamma function ਫਰਮਾ:OEIS2C [1;2,5,1,10,8,1,12,3,1,5,3,5,8,2,1,23,1,2,161,...] 1970 1.46099848620631835815887311784605969 1.92756 19754 82925 30426 Tetranacci constant T {\displaystyle {\mathcal {T}}} Positive root of : x 4 − x 3 − x 2 − x − 1 = 0 {\displaystyle :\;\;x^{4}-x^{3}-x^{2}-x-1=0} A ਫਰਮਾ:OEIS2C [1;1,12,1,4,7,1,21,1,2,1,4,6,1,10,1,2,2,1,7,1,...] 1.92756197548292530426190586173662216 1.00743 47568 84279 37609 DeVicci's tesseract constant f ( 3 , 4 ) {\displaystyle {f_{(3,4)}}} The largest cube that can pass through in an 4D hypercube. Positive root of : 4 x 4 − 28 x 3 − 7 x 2 + 16 x + 16 = 0 {\displaystyle :\;\;4x^{4}{-}28x^{3}{-}7x^{2}{+}16x{+}16=0}
Root [ 4 * x ^ 8-28 * x ^ 6 -7 * x ^ 4 + 16 * x ^ 2 + 16 = 0 ] A ਫਰਮਾ:OEIS2C [1;134,1,1,73,3,1,5,2,1,6,3,11,4,1,5,5,1,1,48,...] 1.00743475688427937609825359523109914 1.70521 11401 05367 76428 Niven's constant C {\displaystyle {C}} 1 + ∑ n = 2 ∞ ( 1 − 1 ζ ( n ) ) {\displaystyle 1+\sum _{n=2}^{\infty }\left(1-{\frac {1}{\zeta (n)}}\right)} 1 + Sum [ n = 2 to ∞ ] { 1 - ( 1 / Zeta ( n ))} ਫਰਮਾ:OEIS2C [1;1,2,2,1,1,4,1,1,3,4,4,8,4,1,1,2,1,1,11,1,...] 1969 1.70521114010536776428855145343450816 0.60459 97880 78072 61686 Relationship among the area of an equilateral triangle and the inscribed circle. π 3 3 {\displaystyle {\frac {\pi }{3{\sqrt {3}}}}} ∑ n = 1 ∞ 1 n ( 2 n n ) = 1 − 1 2 + 1 4 − 1 5 + 1 7 − 1 8 + ⋯ {\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n{2n \choose n}}}=1-{\frac {1}{2}}+{\frac {1}{4}}-{\frac {1}{5}}+{\frac {1}{7}}-{\frac {1}{8}}+\cdots } Dirichlet series Sum [ 1 / ( n Binomial [ 2 n , n ]) , { n , 1 , ∞ }] T ਫਰਮਾ:OEIS2C [0;1,1,1,1,8,10,2,2,3,3,1,9,2,5,4,1,27,27,6,6,...] 0.60459978807807261686469275254738524 1.15470 05383 79251 52901 Hermite constant γ 2 {\displaystyle \gamma _{_{2}}} 2 3 = 1 cos ( π 6 ) {\displaystyle {\frac {2}{\sqrt {3}}}={\frac {1}{\cos \,({\frac {\pi }{6}})}}} A 1+ ਫਰਮਾ:OEIS2C [1;6,2,6,2,6,2,6,2,6,2,6,2,6,2,6,2,6,2,6,2,6,2,...] [1;6,2 ] 1.15470053837925152901829756100391491 0.41245 40336 40107 59778 Prouhet–Thue–Morse constant τ {\displaystyle \tau } ∑ n = 0 ∞ t n 2 n + 1 {\displaystyle \sum _{n=0}^{\infty }{\frac {t_{n}}{2^{n+1}}}} where t n {\displaystyle {t_{n}}} is the Thue–Morse sequence and Where τ ( x ) = ∑ n = 0 ∞ ( − 1 ) t n x n = ∏ n = 0 ∞ ( 1 − x 2 n ) {\displaystyle \tau (x)=\sum _{n=0}^{\infty }(-1)^{t_{n}}\,x^{n}=\prod _{n=0}^{\infty }(1-x^{2^{n}})} T ਫਰਮਾ:OEIS2C [0;2,2,2,1,4,3,5,2,1,4,2,1,5,44,1,4,1,2,4,1,1,...] 0.41245403364010759778336136825845528 0.58057 75582 04892 40229 Pell constant P P e l l {\displaystyle {{\mathcal {P}}_{_{Pell}}}} 1 − ∏ n = 0 ∞ ( 1 − 1 2 2 n + 1 ) {\displaystyle 1-\prod _{n=0}^{\infty }\left(1-{\frac {1}{2^{2n+1}}}\right)} N [ 1 - prod [ n = 0 to ∞ ] { 1-1 / ( 2 ^ ( 2 n + 1 )}] T ? ਫਰਮਾ:OEIS2C [0;1,1,2,1,1,1,1,14,1,3,1,1,6,9,18,7,1,27,1,1,...] 0.58057755820489240229004389229702574 0.66274 34193 49181 58097 Laplace limit λ {\displaystyle {\lambda }} x e x 2 + 1 x 2 + 1 + 1 = 1 {\displaystyle {\frac {x\;e^{\sqrt {x^{2}+1}}}{{\sqrt {x^{2}+1}}+1}}=1} ( x e ^ sqrt ( x ^ 2 + 1 )) / ( sqrt ( x ^ 2 + 1 ) + 1 ) = 1 ਫਰਮਾ:OEIS2C [0;1,1,1,27,1,1,1,8,2,154,2,4,1,5,1,1,2,1601,...] 1782 ~ 0.66274341934918158097474209710925290 0.17150 04931 41536 06586 Hall-Montgomery Constant δ 0 {\displaystyle {{\delta }_{_{0}}}} 1 + π 2 6 + 2 L i 2 ( − e ) L i 2 = Dilogarithm integral {\displaystyle 1+{\frac {\pi ^{2}}{6}}+2\;\mathrm {Li} _{2}\left(-{\sqrt {e}}\;\right)\quad \mathrm {Li} _{2}\,\scriptstyle {\text{= Dilogarithm integral}}} 1 + Pi ^ 2 / 6 + 2 * PolyLog [ 2 , - Sqrt [ E ]] ਫਰਮਾ:OEIS2C [0;5,1,4,1,10,1,1,11,18,1,2,19,14,1,51,1,2,1,...] 0.17150049314153606586043997155521210 1.55138 75245 48320 39226 Calabi triangle constant C C R {\displaystyle {C_{_{CR}}}} 1 3 + ( − 23 + 3 i 237 ) 1 3 3 ⋅ 2 2 3 + 11 3 ( 2 ( − 23 + 3 i 237 ) ) 1 3 {\displaystyle {1 \over 3}+{(-23+3i{\sqrt {237}})^{\tfrac {1}{3}} \over 3\cdot 2^{\tfrac {2}{3}}}+{11 \over 3(2(-23+3i{\sqrt {237}}))^{\tfrac {1}{3}}}} FindRoot [ 2 x ^ 3-2 x ^ 2-3 x + 2 == 0 , { x , 1.5 }, WorkingPrecision -> 40 ] A ਫਰਮਾ:OEIS2C [1;1,1,4,2,1,2,1,5,2,1,3,1,1,390,1,1,2,11,6,2,...] 1946 ~ 1.55138752454832039226195251026462381 1.22541 67024 65177 64512 Gamma(3/4) Γ ( 3 4 ) {\displaystyle \Gamma ({\tfrac {3}{4}})} ( − 1 + 3 4 ) ! = ( − 1 4 ) ! {\displaystyle \left(-1+{\frac {3}{4}}\right)!=\left(-{\frac {1}{4}}\right)!} ਫਰਮਾ:OEIS2C [1;4,2,3,2,2,1,1,1,2,1,4,7,1,171,3,2,3,1,1,8,3,...] 1.22541670246517764512909830336289053 1.20205 69031 59594 28539 Apéry's constant ζ ( 3 ) {\displaystyle \zeta (3)} ∑ n = 1 ∞ 1 n 3 = 1 1 3 + 1 2 3 + 1 3 3 + 1 4 3 + 1 5 3 + ⋯ = {\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n^{3}}}={\frac {1}{1^{3}}}+{\frac {1}{2^{3}}}+{\frac {1}{3^{3}}}+{\frac {1}{4^{3}}}+{\frac {1}{5^{3}}}+\cdots =} 1 2 ∑ n = 1 ∞ H n n 2 = 1 2 ∑ i = 1 ∞ ∑ j = 1 ∞ 1 i j ( i + j ) = ∫ 0 1 ∫ 0 1 ∫ 0 1 d x d y d z 1 − x y z {\displaystyle {\frac {1}{2}}\sum _{n=1}^{\infty }{\frac {H_{n}}{n^{2}}}={\frac {1}{2}}\sum _{i=1}^{\infty }\sum _{j=1}^{\infty }{\frac {1}{ij(i{+}j)}}=\!\!\int \limits _{0}^{1}\!\!\int \limits _{0}^{1}\!\!\int \limits _{0}^{1}{\frac {\mathrm {d} x\mathrm {d} y\mathrm {d} z}{1-xyz}}}
I ਫਰਮਾ:OEIS2C [1;4,1,18,1,1,1,4,1,9,9,2,1,1,1,2,7,1,1,7,11,...] 1979 1.20205690315959428539973816151144999 0.91596 55941 77219 01505 Catalan's constant C {\displaystyle {C}} ∫ 0 1 ∫ 0 1 1 1 + x 2 y 2 d x d y = ∑ n = 0 ∞ ( − 1 ) n ( 2 n + 1 ) 2 = 1 1 2 − 1 3 2 + ⋯ {\displaystyle \int _{0}^{1}\!\!\int _{0}^{1}\!\!{\frac {1}{1{+}x^{2}y^{2}}}\,dx\,dy=\!\sum _{n=0}^{\infty }\!{\frac {(-1)^{n}}{(2n{+}1)^{2}}}\!=\!{\frac {1}{1^{2}}}{-}{\frac {1}{3^{2}}}{+}{\cdots }} Sum [ n = 0 to ∞ ] {( -1 ) ^ n / ( 2 n + 1 ) ^ 2 } T ਫਰਮਾ:OEIS2C [0;1,10,1,8,1,88,4,1,1,7,22,1,2,3,26,1,11,...] 1864 0.91596559417721901505460351493238411 0.78539 81633 97448 30961 Beta(1) β ( 1 ) {\displaystyle {\beta }(1)} π 4 = ∑ n = 0 ∞ ( − 1 ) n 2 n + 1 = 1 1 − 1 3 + 1 5 − 1 7 + 1 9 − ⋯ {\displaystyle {\frac {\pi }{4}}=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{2n+1}}={\frac {1}{1}}-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+{\frac {1}{9}}-\cdots } Sum [ n = 0 to ∞ ] {( -1 ) ^ n / ( 2 n + 1 )} T ਫਰਮਾ:OEIS2C [0; 1,3,1,1,1,15,2,72,1,9,1,17,1,2,1,5,1,1,10,...] 1805 to 1859 0.78539816339744830961566084581987572 0.00131 76411 54853 17810 Heath-Brown–Moroz constant C H B M {\displaystyle {C_{_{HBM}}}} ∏ n = 1 ∞ ( 1 − 1 p n ) 7 ( 1 + 7 p n + 1 p n 2 ) p n : p r i m e {\displaystyle {\underset {p_{n}:\,{prime}}{\prod _{n=1}^{\infty }\left(1-{\frac {1}{p_{n}}}\right)^{7}\left(1+{\frac {7p_{n}+1}{p_{n}^{2}}}\right)}}} N [ prod [ n = 1 to ∞ ] {(( 1-1 / prime ( n )) ^ 7 ) * ( 1 + ( 7 * prime ( n ) + 1 ) / ( prime ( n ) ^ 2 ))}] T ? ਫਰਮਾ:OEIS2C [0;758,1,13,1,2,3,56,8,1,1,1,1,1,143,1,1,1,2,...] 0.00131764115485317810981735232251358 0.56755 51633 06957 82538 Module of ।nfinite Tetration of i | ∞ i | {\displaystyle |{}^{\infty }{i}|} lim n → ∞ | n i | = | lim n → ∞ i i ⋅ ⋅ i ⏟ n | {\displaystyle \lim _{n\to \infty }\left|{}^{n}i\right|=\left|\lim _{n\to \infty }\underbrace {i^{i^{\cdot ^{\cdot ^{i}}}}} _{n}\right|} ਫਰਮਾ:OEIS2C [0;1,1,3,4,1,58,12,1,51,1,4,12,1,1,2,2,3,...] 0.56755516330695782538461314419245334 0.78343 05107 12134 40705 Sophomore's dream1 J.Bernoulli I 1 {\displaystyle {I}_{1}} ∫ 0 1 x x d x = ∑ n = 1 ∞ ( − 1 ) n + 1 n n = 1 1 1 − 1 2 2 + 1 3 3 − ⋯ {\displaystyle \int _{0}^{1}\!x^{x}\,dx=\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n^{n}}}={\frac {1}{1^{1}}}-{\frac {1}{2^{2}}}+{\frac {1}{3^{3}}}-{\cdots }} Sum [ n = 1 to ∞ ] { - ( -1 ) ^ n / n ^ n } ਫਰਮਾ:OEIS2C [0;1,3,1,1,1,1,1,1,2,4,7,2,1,2,1,1,1,2,1,14,...] 1697 0.78343051071213440705926438652697546 1.29128 59970 62663 54040 Sophomore's dream2 J.Bernoulli I 2 {\displaystyle {I}_{2}} ∫ 0 1 1 x x d x = ∑ n = 1 ∞ 1 n n = 1 1 1 + 1 2 2 + 1 3 3 + 1 4 4 + ⋯ {\displaystyle \int _{0}^{1}\!{\frac {1}{x^{x}}}\,dx=\sum _{n=1}^{\infty }{\frac {1}{n^{n}}}={\frac {1}{1^{1}}}+{\frac {1}{2^{2}}}+{\frac {1}{3^{3}}}+{\frac {1}{4^{4}}}+\cdots } ਫਰਮਾ:OEIS2C [1;3,2,3,4,3,1,2,1,1,6,7,2,5,3,1,2,1,8,1,2,4,...] 1697 1.29128599706266354040728259059560054 0.70523 01717 91800 96514 Primorial constant Sum of the product of inverse of primes P # {\displaystyle {P_{\#}}} ∑ n = 1 ∞ 1 p n # = 1 2 + 1 6 + 1 30 + 1 210 + . . . = ∑ k = 1 ∞ ∏ n = 1 k 1 p n p n : p r i m e {\displaystyle {\underset {p_{n}:\,{prime}}{\sum _{n=1}^{\infty }{\frac {1}{p_{n}\#}}={\frac {1}{2}}+{\frac {1}{6}}+{\frac {1}{30}}+{\frac {1}{210}}+...=\sum _{k=1}^{\infty }\prod _{n=1}^{k}{\frac {1}{p_{n}}}}}} Sum [ k = 1 to ∞ ] ( prod [ n = 1 to k ] { 1 / prime ( n )}) I ਫਰਮਾ:OEIS2C [0;1,2,2,1,1,4,1,2,1,1,6,13,1,4,1,16,6,1,1,4,...] 0.70523017179180096514743168288824851 0.14758 36176 50433 27417 Plouffe's gamma constant C {\displaystyle {C}} 1 π arctan 1 2 = 1 π ∑ n = 0 ∞ ( − 1 ) n ( 2 2 n + 1 ) ( 2 n + 1 ) {\displaystyle {\frac {1}{\pi }}\arctan {\frac {1}{2}}={\frac {1}{\pi }}\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2^{2n+1})(2n+1)}}} = 1 π ( 1 2 − 1 3 ⋅ 2 3 + 1 5 ⋅ 2 5 − 1 7 ⋅ 2 7 + ⋯ ) {\displaystyle ={\frac {1}{\pi }}\left({\frac {1}{2}}-{\frac {1}{3\cdot 2^{3}}}+{\frac {1}{5\cdot 2^{5}}}-{\frac {1}{7\cdot 2^{7}}}+\cdots \right)} T ਫਰਮਾ:OEIS2C [0;6,1,3,2,5,1,6,5,3,1,1,2,1,1,2,3,1,2,3,2,2,...] 0.14758361765043327417540107622474052 0.15915 49430 91895 33576 Plouffe's A constant A {\displaystyle {A}} 1 2 π {\displaystyle {\frac {1}{2\pi }}} T ਫਰਮਾ:OEIS2C [0;6,3,1,1,7,2,146,3,6,1,1,2,7,5,5,1,4,1,2,42,...] 0.15915494309189533576888376337251436 0.29156 09040 30818 78013 Dimer constant 2D, Domino tiling C π {\displaystyle {\frac {C}{\pi }}} C=Catalan
∫ − π π cosh − 1 ( cos ( t ) + 3 2 ) 4 π d t {\displaystyle \int \limits _{-\pi }^{\pi }{\frac {\cosh ^{-1}\left({\frac {\sqrt {\cos(t)+3}}{\sqrt {2}}}\right)}{4\pi }}\,dt} N [ int [ - pi to pi ] { arccosh ( sqrt ( cos ( t ) + 3 ) / sqrt ( 2 )) / ( 4 * Pi ) dt }] ਫਰਮਾ:OEIS2C [0;3,2,3,16,8,10,3,1,1,2,1,3,1,2,13,1,1,4,1,5,...] 0.29156090403081878013838445646839491 0.49801 56681 18356 04271 0.15494 98283 01810 68512 i
Factorial(i ) i ! {\displaystyle {i}\,!} Γ ( 1 + i ) = i Γ ( i ) = ∫ 0 ∞ t i e t d t {\displaystyle \Gamma (1+i)=i\,\Gamma (i)=\int \limits _{0}^{\infty }{\frac {t^{i}}{e^{t}}}\mathrm {d} t} C ਫਰਮਾ:OEIS2C ਫਰਮਾ:OEIS2C [0;6,2,4,1,8,1,46,2,2,3,5,1,10,7,5,1,7,2,...] - [0;2,125,2,18,1,2,1,1,19,1,1,1,2,3,34,...] i 0.49801566811835604271369111746219809 - 0.15494982830181068512495513048388 i 2.09455 14815 42326 59148 Wallis Constant W {\displaystyle W} 45 − 1929 18 3 + 45 + 1929 18 3 {\displaystyle {\sqrt[{3}]{\frac {45-{\sqrt {1929}}}{18}}}+{\sqrt[{3}]{\frac {45+{\sqrt {1929}}}{18}}}} ((( 45 - sqrt ( 1929 )) / 18 )) ^ ( 1 / 3 ) + ((( 45 + sqrt ( 1929 )) / 18 )) ^ ( 1 / 3 ) A ਫਰਮਾ:OEIS2C [2;10,1,1,2,1,3,1,1,12,3,5,1,1,2,1,6,1,11,4,...] 1616 to 1703 2.09455148154232659148238654057930296 0.72364 84022 98200 00940 Sarnak constant C s a {\displaystyle {C_{sa}}} ∏ p > 2 ( 1 − p + 2 p 3 ) {\displaystyle \prod _{p>2}{\Big (}1-{\frac {p+2}{p^{3}}}{\Big )}} N [ prod [ k = 2 to ∞ ] { 1 - ( prime ( k ) + 2 ) / ( prime ( k ) ^ 3 )}] T ? ਫਰਮਾ:OEIS2C [0;1,2,1,1,1,1,1,1,1,4,4,1,1,1,1,1,1,1,8,2,1,1,...] 0.72364840229820000940884914980912759 0.63212 05588 28557 67840 Time constant τ {\displaystyle {\tau }} lim n → ∞ 1 − ! n n ! = lim n → ∞ P ( n ) = ∫ 0 1 e − x d x = 1 − 1 e = {\displaystyle \lim _{n\to \infty }1-{\frac {!n}{n!}}=\lim _{n\to \infty }P(n)=\int _{0}^{1}e^{-x}dx=1{-}{\frac {1}{e}}=} ∑ n = 1 ∞ ( − 1 ) n + 1 n ! = 1 1 ! − 1 2 ! + 1 3 ! − 1 4 ! + 1 5 ! − 1 6 ! + ⋯ {\displaystyle \sum \limits _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n!}}={\frac {1}{1!}}{-}{\frac {1}{2!}}{+}{\frac {1}{3!}}{-}{\frac {1}{4!}}{+}{\frac {1}{5!}}{-}{\frac {1}{6!}}{+}\cdots }
lim_ ( n -> ∞ ) ( 1 - ! n / n ! ) ! n = subfactorial T ਫਰਮਾ:OEIS2C [0;1,1,1,2,1,1,4,1,1,6,1,1,8,1,1,10,1,1,12,1,...] = [0;1,1,1,2n ], n∈ℕ 0.63212055882855767840447622983853913 1.04633 50667 70503 18098 Minkowski-Siegel mass constant F 1 {\displaystyle F_{1}} ∏ n = 1 ∞ n ! 2 π n ( n e ) n 1 + 1 n 12 {\displaystyle \prod _{n=1}^{\infty }{\frac {n!}{{\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}{\sqrt[{12}]{1+{\tfrac {1}{n}}}}}}} N [ prod [ n = 1 to ∞ ] n ! / ( sqrt ( 2 * Pi * n ) * ( n / e ) ^ n * ( 1 + 1 / n ) ^ ( 1 / 12 ))] ਫਰਮਾ:OEIS2C [1;21,1,1,2,1,1,4,2,1,5,7,2,1,20,1,1,1134,3,..] 1867 1885 1935 1.04633506677050318098095065697776037 5.24411 51085 84239 62092 Lemniscate Constant 2 ϖ {\displaystyle 2\varpi } [ Γ ( 1 4 ) ] 2 2 π = 4 ∫ 0 1 d x ( 1 − x 2 ) ( 2 − x 2 ) {\displaystyle {\frac {[\Gamma ({\tfrac {1}{4}})]^{2}}{\sqrt {2\pi }}}=4\int _{0}^{1}{\frac {dx}{\sqrt {(1-x^{2})(2-x^{2})}}}} Gamma [ 1 / 4 ] ^ 2 / Sqrt [ 2 Pi ] ਫਰਮਾ:OEIS2C [5;4,10,2,1,2,3,29,4,1,2,1,2,1,2,1,4,9,1,4,1,2,...] 1718 5.24411510858423962092967917978223883 0.66170 71822 67176 23515 Robbins constant Δ ( 3 ) {\displaystyle \Delta (3)} 4 + 17 2 − 6 3 − 7 π 105 + ln ( 1 + 2 ) 5 + 2 ln ( 2 + 3 ) 5 {\displaystyle {\frac {4\!+\!17{\sqrt {2}}\!-6{\sqrt {3}}\!-7\pi }{105}}\!+\!{\frac {\ln(1\!+\!{\sqrt {2}})}{5}}\!+\!{\frac {2\ln(2\!+\!{\sqrt {3}})}{5}}} ( 4 + 17 * 2 ^ ( 1 / 2 ) -6 * 3 ^ ( 1 / 2 ) + 21 * ln ( 1 + 2 ^ ( 1 / 2 )) + 42 * ln ( 2 + 3 ^ ( 1 / 2 )) -7 * Pi ) / 105 ਫਰਮਾ:OEIS2C [0;1,1,1,21,1,2,1,4,10,1,2,2,1,3,11,1,331,1,4,...] 1978 0.66170718226717623515583113324841358 1.30357 72690 34296 39125 Conway constant λ {\displaystyle {\lambda }} x 71 − x 69 − 2 x 68 − x 67 + 2 x 66 + 2 x 65 + x 64 − x 63 − x 62 − x 61 − x 60 − x 59 + 2 x 58 + 5 x 57 + 3 x 56 − 2 x 55 − 10 x 54 − 3 x 53 − 2 x 52 + 6 x 51 + 6 x 50 + x 49 + 9 x 48 − 3 x 47 − 7 x 46 − 8 x 45 − 8 x 44 + 10 x 43 + 6 x 42 + 8 x 41 − 5 x 40 − 12 x 39 + 7 x 38 − 7 x 37 + 7 x 36 + x 35 − 3 x 34 + 10 x 33 + x 32 − 6 x 31 − 2 x 30 − 10 x 29 − 3 x 28 + 2 x 27 + 9 x 26 − 3 x 25 + 14 x 24 − 8 x 23 − 7 x 21 + 9 x 20 + 3 x 19 − 4 x 18 − 10 x 17 − 7 x 16 + 12 x 15 + 7 x 14 + 2 x 13 − 12 x 12 − 4 x 11 − 2 x 10 + 5 x 9 + x 7 − 7 x 6 + 7 x 5 − 4 x 4 + 12 x 3 − 6 x 2 + 3 x − 6 = 0 {\displaystyle {\begin{smallmatrix}x^{71}\quad \ -x^{69}-2x^{68}-x^{67}+2x^{66}+2x^{65}+x^{64}-x^{63}-x^{62}-x^{61}-x^{60}\\-x^{59}+2x^{58}+5x^{57}+3x^{56}-2x^{55}-10x^{54}-3x^{53}-2x^{52}+6x^{51}+6x^{50}\\+x^{49}+9x^{48}-3x^{47}-7x^{46}-8x^{45}-8x^{44}+10x^{43}+6x^{42}+8x^{41}-5x^{40}\\-12x^{39}+7x^{38}-7x^{37}+7x^{36}+x^{35}-3x^{34}+10x^{33}+x^{32}-6x^{31}-2x^{30}\\-10x^{29}-3x^{28}+2x^{27}+9x^{26}-3x^{25}+14x^{24}-8x^{23}\quad \ -7x^{21}+9x^{20}\\+3x^{19}\!-4x^{18}\!-10x^{17}\!-7x^{16}\!+12x^{15}\!+7x^{14}\!+2x^{13}\!-12x^{12}\!-4x^{11}\!-2x^{10}\\+5x^{9}+x^{7}\quad \ -7x^{6}+7x^{5}-4x^{4}+12x^{3}-6x^{2}+3x-6\ =\ 0\quad \quad \quad \end{smallmatrix}}} A ਫਰਮਾ:OEIS2C [1;3,3,2,2,54,5,2,1,16,1,30,1,1,1,2,2,1,14,1,...] 1987 1.30357726903429639125709911215255189 1.18656 91104 15625 45282 Khinchin–Lévy constant β {\displaystyle {\beta }} π 2 12 ln 2 {\displaystyle {\frac {\pi ^{2}}{12\,\ln 2}}} ਫਰਮਾ:OEIS2C [1;5,2,1,3,1,1,28,18,16,3,2,6,2,6,1,1,5,5,9,...] 1935 1.18656911041562545282172297594723712 0.83564 88482 64721 05333 Baker constant β 3 {\displaystyle \beta _{3}} ∫ 0 1 d t 1 + t 3 = ∑ n = 0 ∞ ( − 1 ) n 3 n + 1 = 1 3 ( ln 2 + π 3 ) {\displaystyle \int _{0}^{1}{\frac {\mathrm {d} t}{1+t^{3}}}=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{3n+1}}={\frac {1}{3}}\left(\ln 2+{\frac {\pi }{\sqrt {3}}}\right)} Sum [ n = 0 to ∞ ] {(( -1 ) ^ ( n )) / ( 3 n + 1 )} ਫਰਮਾ:OEIS2C [0;1,5,11,1,4,1,6,1,4,1,1,1,2,1,3,2,2,2,2,1,3,...] 0.83564884826472105333710345970011076 23.10344 79094 20541 6160 Kempner Serie (0) K 0 {\displaystyle {K_{0}}} 1 + 1 2 + 1 3 + ⋯ + 1 9 + 1 11 + ⋯ + 1 19 + 1 21 + ⋯ {\displaystyle 1{+}{\frac {1}{2}}{+}{\frac {1}{3}}{+}\cdots {+}{\frac {1}{9}}{+}{\frac {1}{11}}{+}\cdots {+}{\frac {1}{19}}{+}{\frac {1}{21}}{+}\cdots } + 1 99 + 1 111 + ⋯ + 1 119 + 1 121 + ⋯ {\displaystyle {+}{\frac {1}{99}}{+}{\frac {1}{111}}{+}\cdots {+}{\frac {1}{119}}{+}{\frac {1}{121}}{+}\cdots }
(Excluding all denominators containing 0.)
1 + 1 / 2 + 1 / 3 + 1 / 4 + 1 / 5 + 1 / 6 + 1 / 7 + 1 / 8 + 1 / 9 + 1 / 11 + 1 / 12 + 1 / 13 + 1 / 14 + 1 / 15 + ... ਫਰਮਾ:OEIS2C [23;9,1,2,3244,1,1,5,1,2,2,8,3,1,1,6,1,84,1,...] 23.1034479094205416160340540433255981 0.98943 12738 31146 95174 Lebesgue constant C 1 {\displaystyle {C_{1}}} lim n → ∞ ( L n − 4 π 2 ln ( 2 n + 1 ) ) = 4 π 2 ( ∑ k = 1 ∞ 2 ln k 4 k 2 − 1 − Γ ′ ( 1 2 ) Γ ( 1 2 ) ) {\displaystyle \lim _{n\to \infty }\!\!\left(\!{L_{n}{-}{\frac {4}{\pi ^{2}}}\ln(2n{+}1)}\!\!\right)\!{=}{\frac {4}{\pi ^{2}}}\!\left({\sum _{k=1}^{\infty }\!{\frac {2\ln k}{4k^{2}{-}1}}}{-}{\frac {\Gamma '({\tfrac {1}{2}})}{\Gamma ({\tfrac {1}{2}})}}\!\!\right)} 4 / pi ^ 2 * [( 2 Sum [ k = 1 to ∞ ] { ln ( k ) / ( 4 * k ^ 2-1 )}) - poligamma ( 1 / 2 )] ਫਰਮਾ:OEIS2C [0;1,93,1,1,1,1,1,1,1,7,1,12,2,15,1,2,7,2,1,5,...] ? 0.98943127383114695174164880901886671 0.19452 80494 65325 11361 2nd du Bois-Reymond constant C 2 {\displaystyle {C_{2}}} e 2 − 7 2 = ∫ 0 ∞ | d d t ( sin t t ) n | d t − 1 {\displaystyle {\frac {e^{2}-7}{2}}=\int _{0}^{\infty }\left|{{\frac {d}{dt}}\left({\frac {\sin t}{t}}\right)^{n}}\right|\,dt-1} T ਫਰਮਾ:OEIS2C [0;5,7,9,11,13,15,17,19,21,23,25,27,29,31,...] = [0;2p+3 ], p∈ℕ 0.19452804946532511361521373028750390 0.78853 05659 11508 96106 Lüroth constant C L {\displaystyle C_{L}} ∑ n = 2 ∞ ln ( n n − 1 ) n {\displaystyle \sum _{n=2}^{\infty }{\frac {\ln \left({\frac {n}{n-1}}\right)}{n}}} Sum [ n = 2 to ∞ ] log ( n / ( n -1 )) / n ਫਰਮਾ:OEIS2C [0;1,3,1,2,1,2,4,1,127,1,2,2,1,3,8,1,1,2,1,16,...] 0.78853056591150896106027632216944432 1.18745 23511 26501 05459 Foias constant α F α {\displaystyle F_{\alpha }} x n + 1 = ( 1 + 1 x n ) n for n = 1 , 2 , 3 , … {\displaystyle x_{n+1}=\left(1+{\frac {1}{x_{n}}}\right)^{n}{\text{ for }}n=1,2,3,\ldots } Foias constant is the unique real number such that if x 1 = α then the sequence diverges to ∞. When x 1 = α , lim n → ∞ x n log n n = 1 {\displaystyle \,\lim _{n\to \infty }x_{n}{\tfrac {\log n}{n}}=1}
ਫਰਮਾ:OEIS2C [1;5,2,1,81,3,2,2,1,1,1,1,1,6,1,1,3,1,1,4,3,2,...] 2000 1.18745235112650105459548015839651935 2.29316 62874 11861 03150 Foias constant β F β {\displaystyle F_{\beta }} x x + 1 = ( x + 1 ) x {\displaystyle x^{x+1}=(x+1)^{x}} ਫਰਮਾ:OEIS2C [2;3,2,2,3,4,2,3,2,130,1,1,1,1,1,6,3,2,1,15,1,...] 2000 2.29316628741186103150802829125080586 0.82246 70334 24113 21823 Nielsen-Ramanujan constant ζ ( 2 ) 2 {\displaystyle {\frac {{\zeta }(2)}{2}}} π 2 12 = ∑ n = 1 ∞ ( − 1 ) n + 1 n 2 = 1 1 2 − 1 2 2 + 1 3 2 − 1 4 2 + 1 5 2 − ⋯ {\displaystyle {\frac {\pi ^{2}}{12}}=\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n^{2}}}={\frac {1}{1^{2}}}{-}{\frac {1}{2^{2}}}{+}{\frac {1}{3^{2}}}{-}{\frac {1}{4^{2}}}{+}{\frac {1}{5^{2}}}{-}\cdots } Sum [ n = 1 to ∞ ] {(( -1 ) ^ ( n + 1 )) / n ^ 2 } T ਫਰਮਾ:OEIS2C [0;1,4,1,1,1,2,1,1,1,1,3,2,2,4,1,1,1,1,1,1,4...] 1909 0.82246703342411321823620758332301259 0.69314 71805 59945 30941 Natural logarithm of 2 L n ( 2 ) {\displaystyle Ln(2)} ∑ n = 1 ∞ 1 n 2 n = ∑ n = 1 ∞ ( − 1 ) n + 1 n = 1 1 − 1 2 + 1 3 − 1 4 + ⋯ {\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n2^{n}}}=\sum _{n=1}^{\infty }{\frac {({-}1)^{n+1}}{n}}={\frac {1}{1}}-{\frac {1}{2}}+{\frac {1}{3}}-{\frac {1}{4}}+{\cdots }} Sum [ n = 1 to ∞ ] {( -1 ) ^ ( n + 1 ) / n } T ਫਰਮਾ:OEIS2C [0;1,2,3,1,6,3,1,1,2,1,1,1,1,3,10,1,1,1,2,1,1,...] 1550 to 1617 0.69314718055994530941723212145817657 0.47494 93799 87920 65033 Weierstrass constant σ ( 1 2 ) {\displaystyle \sigma ({\tfrac {1}{2}})} e π 8 π 4 ⋅ 2 3 / 4 ( 1 4 ! ) 2 {\displaystyle {\frac {e^{\frac {\pi }{8}}{\sqrt {\pi }}}{4\cdot 2^{3/4}{({\frac {1}{4}}!)^{2}}}}} ( E ^ ( Pi / 8 ) Sqrt [ Pi ]) / ( 4 2 ^ ( 3 / 4 ) ( 1 / 4 ) !^ 2 ) ਫਰਮਾ:OEIS2C [0;2,9,2,11,1,6,1,4,6,3,19,9,217,1,2,4,8,6...] 1872 ? 0.47494937998792065033250463632798297 0.57721 56649 01532 86060 Euler–Mascheroni constant γ {\displaystyle {\gamma }} ∑ n = 1 ∞ ∑ k = 0 ∞ ( − 1 ) k 2 n + k = ∑ n = 1 ∞ ( 1 n − ln ( 1 + 1 n ) ) {\displaystyle \sum _{n=1}^{\infty }\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{2^{n}+k}}=\sum _{n=1}^{\infty }\left({\frac {1}{n}}-\ln \left(1+{\frac {1}{n}}\right)\right)} = ∫ 0 1 − ln ( ln 1 x ) d x = − Γ ′ ( 1 ) = − Ψ ( 1 ) {\displaystyle =\int _{0}^{1}-\ln \left(\ln {\frac {1}{x}}\right)\,dx=-\Gamma '(1)=-\Psi (1)}
sum [ n = 1 to ∞ ] | sum [ k = 0 to ∞ ] {(( -1 ) ^ k ) / ( 2 ^ n + k )} ਫਰਮਾ:OEIS2C [0;1,1,2,1,2,1,4,3,13,5,1,1,8,1,2,4,1,1,40,1,...] 1735 0.57721566490153286060651209008240243 1.38135 64445 18497 79337 Beta, Kneser-Mahler polynomial constant β {\displaystyle \beta } e 2 π ∫ 0 π 3 t tan t d t = e ∫ − 1 3 1 3 ln ⌊ 1 + e 2 π i t ⌋ d t {\displaystyle e^{^{\textstyle {\frac {2}{\pi }}\displaystyle {\int _{0}^{\frac {\pi }{3}}}\textstyle {t\tan t\ dt}}}=e^{^{\displaystyle {\,\int _{\frac {-1}{3}}^{\frac {1}{3}}}\textstyle {\,\ln \lfloor 1+e^{2\pi it}}\rfloor dt}}} e ^ (( PolyGamma ( 1 , 4 / 3 ) - PolyGamma ( 1 , 2 / 3 ) + 9 ) / ( 4 * sqrt ( 3 ) * Pi )) ਫਰਮਾ:OEIS2C [1;2,1,1,1,1,1,4,1,139,2,1,3,5,16,2,1,1,7,2,1,...] 1963 1.38135644451849779337146695685062412 1.35845 62741 82988 43520 Golden Spiral c {\displaystyle c} φ 2 π = ( 1 + 5 2 ) 2 π {\displaystyle \varphi ^{\frac {2}{\pi }}=\left({\frac {1+{\sqrt {5}}}{2}}\right)^{\frac {2}{\pi }}} ਫਰਮਾ:OEIS2C [1;2,1,3,1,3,10,8,1,1,8,1,15,6,1,3,1,1,2,3,1,1,...] 1.35845627418298843520618060050187945 0.57595 99688 92945 43964 Stephens constant C S {\displaystyle C_{S}} ∏ n = 1 ∞ ( 1 − p p 3 − 1 ) {\displaystyle \prod _{n=1}^{\infty }\left(1-{\frac {p}{p^{3}-1}}\right)} Prod [ n = 1 to ∞ ] { 1 - hprime ( n ) / ( hprime ( n ) ^ 3-1 )} T ? ਫਰਮਾ:OEIS2C [0;1,1,2,1,3,1,3,1,2,1,77,2,1,1,10,2,1,1,1,7,...] ? 0.57595996889294543964316337549249669 0.73908 51332 15160 64165 Dottie number d {\displaystyle d} lim x → ∞ cos [ x ] ( c ) = lim x → ∞ cos ( cos ( cos ( ⋯ ( cos ( c ) ) ) ) ) ⏟ x {\displaystyle \lim _{x\to \infty }\cos ^{[x]}(c)=\lim _{x\to \infty }\underbrace {\cos(\cos(\cos(\cdots (\cos(c)))))} _{x}} T ਫਰਮਾ:OEIS2C [0;1,2,1,4,1,40,1,9,4,2,1,15,2,12,1,21,1,17,...] ? 0.73908513321516064165531208767387340 0.67823 44919 17391 97803 Taniguchi constant C T {\displaystyle C_{T}} ∏ n = 1 ∞ ( 1 − 3 p n 3 + 2 p n 4 + 1 p n 5 − 1 p n 6 ) {\displaystyle \prod _{n=1}^{\infty }\left(1-{\frac {3}{{p_{n}}^{3}}}+{\frac {2}{{p_{n}}^{4}}}+{\frac {1}{{p_{n}}^{5}}}-{\frac {1}{{p_{n}}^{6}}}\right)} p n = prime {\displaystyle \scriptstyle p_{n}=\,{\text{prime}}} Prod [ n = 1 to ∞ ] { 1 -3 / ithprime ( n ) ^ 3 + 2 / ithprime ( n ) ^ 4 + 1 / ithprime ( n ) ^ 5 -1 / ithprime ( n ) ^ 6 } T ? ਫਰਮਾ:OEIS2C [0;1,2,9,3,1,2,9,11,1,13,2,15,1,1,1,2,4,1,1,1,...] ? 0.67823449191739197803553827948289481 1.85407 46773 01371 91843 Gauss' Lemniscate constant L / 2 {\displaystyle L{\text{/}}{\sqrt {2}}} ∫ 0 ∞ d x 1 + x 4 = 1 4 π Γ ( 1 4 ) 2 = 4 ( 1 4 ! ) 2 π {\displaystyle \int \limits _{0}^{\infty }{\frac {\mathrm {d} x}{\sqrt {1+x^{4}}}}={\frac {1}{4{\sqrt {\pi }}}}\,\Gamma \left({\frac {1}{4}}\right)^{2}={\frac {4\left({\frac {1}{4}}!\right)^{2}}{\sqrt {\pi }}}} Γ ( ) = Gamma function {\displaystyle \scriptstyle \Gamma (){\text{= Gamma function}}} pi ^ ( 3 / 2 ) / ( 2 Gamma ( 3 / 4 ) ^ 2 ) ਫਰਮਾ:OEIS2C [1;1,5,1,5,1,3,1,6,2,1,4,16,3,112,2,1,1,18,1,...] 1.85407467730137191843385034719526005 1.75874 36279 51184 82469 Infinite product constant, with Alladi-Grinstead P r 1 {\displaystyle Pr_{1}} ∏ n = 2 ∞ ( 1 + 1 n ) 1 n {\displaystyle \prod _{n=2}^{\infty }{\Big (}1+{\frac {1}{n}}{\Big )}^{\frac {1}{n}}} Prod [ n = 2 to inf ] {( 1 + 1 / n ) ^ ( 1 / n )} ਫਰਮਾ:OEIS2C [1;1,3,6,1,8,1,4,3,1,4,1,1,1,6,5,2,40,1,387,2,...] 1977 1.75874362795118482469989684865589317 1.86002 50792 21190 30718 Spiral of Theodorus ∂ {\displaystyle \partial } ∑ n = 1 ∞ 1 n 3 + n = ∑ n = 1 ∞ 1 n ( n + 1 ) {\displaystyle \sum _{n=1}^{\infty }{\frac {1}{{\sqrt {n^{3}}}+{\sqrt {n}}}}=\sum _{n=1}^{\infty }{\frac {1}{{\sqrt {n}}(n+1)}}} Sum [ n = 1 to ∞ ] { 1 / ( n ^ ( 3 / 2 ) + n ^ ( 1 / 2 ))} ਫਰਮਾ:OEIS2C [1;1,6,6,1,15,11,5,1,1,1,1,5,3,3,3,2,1,1,2,19,...] -460 to -399 1.86002507922119030718069591571714332 2.79128 78474 77920 00329 Nested radical S5 S 5 {\displaystyle S_{5}} 21 + 1 2 = 5 + 5 + 5 + 5 + 5 + ⋯ {\displaystyle \displaystyle {\frac {{\sqrt {21}}+1}{2}}=\scriptstyle \,{\sqrt {5+{\sqrt {5+{\sqrt {5+{\sqrt {5+{\sqrt {5+\cdots }}}}}}}}}}} = 1 + 5 − 5 − 5 − 5 − 5 − ⋯ {\displaystyle =1+\,\scriptstyle {\sqrt {5-{\sqrt {5-{\sqrt {5-{\sqrt {5-{\sqrt {5-\cdots }}}}}}}}}}}
A A222134 [2;1,3,1,3,1,3,1,3,1,3,1,3,1,3,1,3,1,3,1,3,1,3,...] [2;1,3 ] ? 2.79128784747792000329402359686400424 0.70710 67811 86547 52440 +0.70710 67811 86547 524 i Square root of i i {\displaystyle {\sqrt {i}}} − 1 4 = 1 + i 2 = e i π 4 = cos ( π 4 ) + i sin ( π 4 ) {\displaystyle {\sqrt[{4}]{-1}}={\frac {1+i}{\sqrt {2}}}=e^{\frac {i\pi }{4}}=\cos \left({\frac {\pi }{4}}\right)+i\sin \left({\frac {\pi }{4}}\right)} C A ਫਰਮਾ:OEIS2C [0;1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,..] = [0;1,2 ,...] [0;1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,..] i = [0;1,2 ,...] i ? 0.70710678118654752440084436210484903 + 0.70710678118654752440084436210484 i 0.80939 40205 40639 13071 Alladi–Grinstead constant A A G {\displaystyle {{\mathcal {A}}_{AG}}} e − 1 + ∑ k = 2 ∞ ∑ n = 1 ∞ 1 n k n + 1 = e − 1 − ∑ k = 2 ∞ 1 k ln ( 1 − 1 k ) {\displaystyle e^{-1+\sum \limits _{k=2}^{\infty }\sum \limits _{n=1}^{\infty }{\frac {1}{nk^{n+1}}}}=e^{-1-\sum \limits _{k=2}^{\infty }{\frac {1}{k}}\ln \left(1-{\frac {1}{k}}\right)}} e ^ {( sum [ k = 2 to ∞ ] | sum [ n = 1 to ∞ ] { 1 / ( n k ^ ( n + 1 ))}) -1 } ਫਰਮਾ:OEIS2C [0;1,4,4,17,4,3,2,5,3,1,1,1,1,6,1,1,2,1,22,...] 1977 0.80939402054063913071793188059409131 2.58498 17595 79253 21706 Sierpiński's constant K {\displaystyle {K}} π ( 2 γ + ln 4 π 3 Γ ( 1 4 ) 4 ) = π ( 2 γ + 4 ln Γ ( 3 4 ) − ln π ) {\displaystyle \pi \left(2\gamma +\ln {\frac {4\pi ^{3}}{\Gamma ({\tfrac {1}{4}})^{4}}}\right)=\pi (2\gamma +4\ln \Gamma ({\tfrac {3}{4}})-\ln \pi )} = π ( 2 ln 2 + 3 ln π + 2 γ − 4 ln Γ ( 1 4 ) ) {\displaystyle =\pi \left(2\ln 2+3\ln \pi +2\gamma -4\ln \Gamma ({\tfrac {1}{4}})\right)}
- Pi Log [ Pi ] + 2 Pi EulerGamma + 4 Pi Log [ Gamma [ 3 / 4 ]] ਫਰਮਾ:OEIS2C [2;1,1,2,2,3,1,3,1,9,2,8,4,1,13,3,1,15,18,1,...] 1907 2.58498175957925321706589358738317116 1.73245 47146 00633 47358 Reciprocal of the Euler–Mascheroni constant 1 γ {\displaystyle {\frac {1}{\gamma }}} ( ∫ 0 1 − log ( log 1 x ) d x ) − 1 = ∑ n = 1 ∞ ( − 1 ) n ( − 1 + γ ) n {\displaystyle \left(\int _{0}^{1}-\log \left(\log {\frac {1}{x}}\right)\,dx\right)^{-1}=\sum _{n=1}^{\infty }(-1)^{n}(-1+\gamma )^{n}} 1 / Integrate_ { x = 0 to 1 } - log ( log ( 1 / x )) ਫਰਮਾ:OEIS2C [1;1,2,1,2,1,4,3,13,5,1,1,8,1,2,4,1,1,40,1,11,...] 1.73245471460063347358302531586082968 1.43599 11241 76917 43235 Lebesgue constant (interpolation) L 1 {\displaystyle {L_{1}}} ∏ i = 0 j ≠ i n x − x i x j − x i = 1 π ∫ 0 π ⌊ sin 3 t 2 ⌋ sin t 2 d t = 1 3 + 2 3 π {\displaystyle \prod _{\begin{smallmatrix}i=0\\j\neq i\end{smallmatrix}}^{n}{\frac {x-x_{i}}{x_{j}-x_{i}}}={\frac {1}{\pi }}\int _{0}^{\pi }{\frac {\lfloor \sin {\frac {3t}{2}}\rfloor }{\sin {\frac {t}{2}}}}\,dt={\frac {1}{3}}+{\frac {2{\sqrt {3}}}{\pi }}} T ਫਰਮਾ:OEIS2C [1;2,3,2,2,6,1,1,1,1,4,1,7,1,1,1,2,1,3,1,2,1,1,...] 1902 ~ 1.43599112417691743235598632995927221 3.24697 96037 17467 06105 Silver root Tutte–Beraha constant ς {\displaystyle \varsigma } 2 + 2 cos 2 π 7 = 2 + 2 + 7 + 7 7 + 7 7 + ⋯ 3 3 3 1 + 7 + 7 7 + 7 7 + ⋯ 3 3 3 {\displaystyle 2+2\cos {\frac {2\pi }{7}}=\textstyle 2+{\frac {2+{\sqrt[{3}]{7+7{\sqrt[{3}]{7+7{\sqrt[{3}]{\,7+\cdots }}}}}}}{1+{\sqrt[{3}]{7+7{\sqrt[{3}]{7+7{\sqrt[{3}]{\,7+\cdots }}}}}}}}} A ਫਰਮਾ:OEIS2C [3;4,20,2,3,1,6,10,5,2,2,1,2,2,1,18,1,1,3,2,...] 3.24697960371746706105000976800847962 1.94359 64368 20759 20505 Euler Totient constant E T {\displaystyle ET} ∏ p ( 1 + 1 p ( p − 1 ) ) p = primes = ζ ( 2 ) ζ ( 3 ) ζ ( 6 ) = 315 ζ ( 3 ) 2 π 4 {\displaystyle {\underset {p{\text{= primes}}}{\prod _{p}{\Big (}1+{\frac {1}{p(p-1)}}{\Big )}}}={\frac {\zeta (2)\zeta (3)}{\zeta (6)}}={\frac {315\zeta (3)}{2\pi ^{4}}}} ਫਰਮਾ:OEIS2C [1;1,16,1,2,1,2,3,1,1,3,2,1,8,1,1,1,1,1,1,1,32,...] 1750 1.94359643682075920505707036257476343 1.49534 87812 21220 54191 Fourth root of five 5 4 {\displaystyle {\sqrt[{4}]{5}}} 5 5 5 5 5 ⋯ 5 5 5 5 5 {\displaystyle {\sqrt[{5}]{5\,{\sqrt[{5}]{5\,{\sqrt[{5}]{5\,{\sqrt[{5}]{5\,{\sqrt[{5}]{5\,\cdots }}}}}}}}}}} ( 5 ( 5 ( 5 ( 5 ( 5 ( 5 ( 5 ) ^ 1 / 5 ) ^ 1 / 5 ) ^ 1 / 5 ) ^ 1 / 5 ) ^ 1 / 5 ) ^ 1 / 5 ) ^ 1 / 5 ... A ਫਰਮਾ:OEIS2C [1;2,53,4,96,2,1,6,2,2,2,6,1,4,1,49,17,2,3,2,...] 1.49534878122122054191189899414091339 0.87228 40410 65627 97617 Area of Ford circle A C F {\displaystyle A_{CF}} ∑ q ≥ 1 ∑ ( p , q ) = 1 1 ≤ p < q π ( 1 2 q 2 ) 2 = π 4 ζ ( 3 ) ζ ( 4 ) = 45 2 ζ ( 3 ) π 3 ζ ( ) = Riemann Zeta Function {\displaystyle \sum _{q\geq 1}\sum _{(p,q)=1 \atop 1\leq p [0;1,6,1,4,1,7,5,36,3,29,1,1,10,3,2,8,1,1,1,3,...] 0.87228404106562797617519753217122587 1.08232 32337 11138 19151 Zeta(4) ζ ( 4 ) {\displaystyle \zeta (4)} π 4 90 = ∑ n = 1 ∞ 1 n 4 = 1 1 4 + 1 2 4 + 1 3 4 + 1 4 4 + 1 5 4 + . . . {\displaystyle {\frac {\pi ^{4}}{90}}=\sum _{n=1}^{\infty }{\frac {1}{n^{4}}}={\frac {1}{1^{4}}}+{\frac {1}{2^{4}}}+{\frac {1}{3^{4}}}+{\frac {1}{4^{4}}}+{\frac {1}{5^{4}}}+...} T ਫਰਮਾ:OEIS2C [1;12,6,1,3,1,4,183,1,1,2,1,3,1,1,5,4,2,7,23,...] ? 1.08232323371113819151600369654116790 1.56155 28128 08830 27491 Triangular root of 2. R 2 {\displaystyle {R_{2}}} 17 − 1 2 = 4 + 4 + 4 + 4 + 4 + 4 + ⋯ − 1 {\displaystyle {\frac {{\sqrt {17}}-1}{2}}=\,\scriptstyle {\sqrt {4+{\sqrt {4+{\sqrt {4+{\sqrt {4+{\sqrt {4+{\sqrt {4+\cdots }}}}}}}}}}}}\,\,-1} = 4 − 4 − 4 − 4 − 4 − 4 − ⋯ {\displaystyle =\,\scriptstyle {\sqrt {4-{\sqrt {4-{\sqrt {4-{\sqrt {4-{\sqrt {4-{\sqrt {4-\cdots }}}}}}}}}}}}\textstyle }
A ਫਰਮਾ:OEIS2C [1;1,1,3,1,1,3,1,1,3,1,1,3,1,1,3,1,1,3,1,1,3,1,...] [1;1,1,3 ] 1.56155281280883027491070492798703851 9.86960 44010 89358 61883 Pi Squared π 2 {\displaystyle {\pi }^{2}} 6 ζ ( 2 ) = 6 ∑ n = 1 ∞ 1 n 2 = 6 1 2 + 6 2 2 + 6 3 2 + 6 4 2 + ⋯ {\displaystyle 6\,\zeta (2)=6\sum _{n=1}^{\infty }{\frac {1}{n^{2}}}={\frac {6}{1^{2}}}+{\frac {6}{2^{2}}}+{\frac {6}{3^{2}}}+{\frac {6}{4^{2}}}+\cdots } T A002388 [9;1,6,1,2,47,1,8,1,1,2,2,1,1,8,3,1,10,5,1,3,...] 9.86960440108935861883449099987615114 1.32471 79572 44746 02596 Plastic number ρ {\displaystyle {\rho }} 1 + 1 + 1 + ⋯ 3 3 3 = 1 2 + 23 108 3 + 1 2 − 23 108 3 {\displaystyle {\sqrt[{3}]{1+\!{\sqrt[{3}]{1+\!{\sqrt[{3}]{1+\cdots }}}}}}=\textstyle {\sqrt[{3}]{{\frac {1}{2}}+\!{\sqrt {\frac {23}{108}}}}}+\!{\sqrt[{3}]{{\frac {1}{2}}-\!{\sqrt {\frac {23}{108}}}}}} ( 1 + ( 1 + ( 1 + ( 1 + ( 1 + ( 1 ) ^ ( 1 / 3 )) ^ ( 1 / 3 )) ^ ( 1 / 3 )) ^ ( 1 / 3 )) ^ ( 1 / 3 )) ^ ( 1 / 3 ) A ਫਰਮਾ:OEIS2C [1;3,12,1,1,3,2,3,2,4,2,141,80,2,5,1,2,8,2,...] 1929 1.32471795724474602596090885447809734 2.37313 82208 31250 90564 Lévy 2 constant 2 l n γ {\displaystyle 2\,ln\,\gamma } π 2 6 l n ( 2 ) {\displaystyle {\frac {\pi ^{2}}{6ln(2)}}} T ਫਰਮਾ:OEIS2C [2;2,1,2,8,57,9,32,1,1,2,1,2,1,2,1,2,1,3,2,...] 1936 2.37313822083125090564344595189447424 0.85073 61882 01867 26036 Regular paperfolding sequence P f {\displaystyle {P_{f}}} ∑ n = 0 ∞ 8 2 n 2 2 n + 2 − 1 = ∑ n = 0 ∞ 1 2 2 n 1 − 1 2 2 n + 2 {\displaystyle \sum _{n=0}^{\infty }{\frac {8^{2^{n}}}{2^{2^{n+2}}-1}}=\sum _{n=0}^{\infty }{\cfrac {\tfrac {1}{2^{2^{n}}}}{1-{\tfrac {1}{2^{2^{n+2}}}}}}} N [ Sum [ n = 0 to ∞ ] { 8 ^ 2 ^ n / ( 2 ^ 2 ^ ( n + 2 ) -1 )}, 37 ] ਫਰਮਾ:OEIS2C [0;1,5,1,2,3,21,1,4,107,7,5,2,1,2,1,1,2,1,6,...] 0.85073618820186726036779776053206660 1.15636 26843 32269 71685 Cubic recurrence constant{{.}} σ 3 {\displaystyle {\sigma _{3}}} ∏ n = 1 ∞ n 3 − n = 1 2 3 ⋯ 3 3 3 = 1 1 / 3 2 1 / 9 3 1 / 27 ⋯ {\displaystyle \prod _{n=1}^{\infty }n^{{3}^{-n}}={\sqrt[{3}]{1{\sqrt[{3}]{2{\sqrt[{3}]{3\cdots }}}}}}=1^{1/3}\;2^{1/9}\;3^{1/27}\cdots } prod [ n = 1 to ∞ ] { n ^ ( 1 / 3 ) ^ n } ਫਰਮਾ:OEIS2C [1;6,2,1,1,8,13,1,3,2,2,6,2,1,2,1,1,1,10,33,...] 1.15636268433226971685337032288736935 1.26185 95071 42914 87419 Fractal dimension of the Koch snowflake C k {\displaystyle {C_{k}}} log 4 log 3 {\displaystyle {\frac {\log 4}{\log 3}}} T A100831 [1;3,1,4,1,1,11,1,46,1,5,112,1,1,1,1,1,3,1,7,...] 1.26185950714291487419905422868552171 6.58088 59910 17920 97085 Froda constant 2 e {\displaystyle 2^{\,e}} 2 e {\displaystyle 2^{e}} [6;1,1,2,1,1,2,3,1,14,11,4,3,1,1,7,5,5,2,7,...] 6.58088599101792097085154240388648649 0.26149 72128 47642 78375 Meissel-Mertens constant M {\displaystyle {M}} lim n → ∞ ( ∑ p ≤ n 1 p − ln ( ln ( n ) ) ) = γ + ∑ p ( ln ( 1 − 1 p ) + 1 p ) γ : Euler constant , p : prime {\displaystyle \lim _{n\rightarrow \infty }\!\!\left(\sum _{p\leq n}{\frac {1}{p}}\!-\ln(\ln(n))\!\right)\!\!={\underset {\!\!\!\!\gamma :\,{\text{Euler constant}},\,\,p:\,{\text{prime}}}{\!\gamma \!+\!\!\sum _{p}\!\left(\!\ln \!\left(\!1\!-\!{\frac {1}{p}}\!\right)\!\!+\!{\frac {1}{p}}\!\right)}}} gamma + Sum [ n = 1 to ∞ ] { ln ( 1-1 / prime ( n )) + 1 / prime ( n )} T ? ਫਰਮਾ:OEIS2C [0;3,1,4,1,2,5,2,1,1,1,1,13,4,2,4,2,1,33,296,...] 1866 & 1873 0.26149721284764278375542683860869585 4.81047 73809 65351 65547 John constant γ {\displaystyle \gamma } i i = i − i = ( i i ) − 1 = ( ( ( i ) i ) i ) i = e π 2 = ∑ n = 0 ∞ π n n ! {\displaystyle {\sqrt[{i}]{i}}=i^{-i}=(i^{i})^{-1}=(((i)^{i})^{i})^{i}=e^{\frac {\pi }{2}}={\sqrt {\sum _{n=0}^{\infty }{\frac {\pi ^{n}}{n!}}}}} T ਫਰਮਾ:OEIS2C [4;1,4,3,1,1,1,1,1,1,1,1,7,1,20,1,3,6,10,3,2,...] 4.81047738096535165547303566670383313 -0.5 ± 0.86602 54037 84438 64676 i Cube Root of 1 1 3 {\displaystyle {\sqrt[{3}]{1}}} { 1 − 1 2 + 3 2 i − 1 2 − 3 2 i . {\displaystyle {\begin{cases}\ \ 1\\-{\frac {1}{2}}+{\frac {\sqrt {3}}{2}}i\\-{\frac {1}{2}}-{\frac {\sqrt {3}}{2}}i.\end{cases}}} 1 , E ^ ( 2 i pi / 3 ), E ^ ( -2 i pi / 3 ) C A ਫਰਮਾ:OEIS2C - [0,5] ± [0;1,6,2,6,2,6,2,6,2,6,2,6,2,6,2,6,2,6,2,...] i - [0,5] ± [0; 1, 6, 2 ] i - 0.5 ± 0.8660254037844386467637231707529 i 0.11000 10000 00000 00000 0001 Liouville number £ L i {\displaystyle {\text{£}}_{Li}} ∑ n = 1 ∞ 1 10 n ! = 1 10 1 ! + 1 10 2 ! + 1 10 3 ! + 1 10 4 ! + ⋯ {\displaystyle \sum _{n=1}^{\infty }{\frac {1}{10^{n!}}}={\frac {1}{10^{1!}}}+{\frac {1}{10^{2!}}}+{\frac {1}{10^{3!}}}+{\frac {1}{10^{4!}}}+\cdots } T ਫਰਮਾ:OEIS2C [1;9,1,999,10,9999999999999,1,9,999,1,9] 0.11000100000000000000000100... 0.06598 80358 45312 53707 Lower limit of Tetration e − e {\displaystyle {e}^{-e}} ( 1 e ) e {\displaystyle \left({\frac {1}{e}}\right)^{e}} ਫਰਮਾ:OEIS2C [0;15,6,2,13,1,3,6,2,1,1,5,1,1,1,9,4,1,1,1,...] 0.06598803584531253707679018759684642 1.83928 67552 14161 13255 Tribonacci constant ϕ 3 {\displaystyle {\phi _{}}_{3}} 1 + 19 + 3 33 3 + 19 − 3 33 3 3 = 1 + ( 1 2 + 1 2 + 1 2 + . . . 3 3 3 ) − 1 {\displaystyle \textstyle {\frac {1+{\sqrt[{3}]{19+3{\sqrt {33}}}}+{\sqrt[{3}]{19-3{\sqrt {33}}}}}{3}}=\scriptstyle \,1+\left({\sqrt[{3}]{{\tfrac {1}{2}}+{\sqrt[{3}]{{\tfrac {1}{2}}+{\sqrt[{3}]{{\tfrac {1}{2}}+...}}}}}}\right)^{-1}} ( 1 / 3 ) * ( 1 + ( 19 + 3 * sqrt ( 33 )) ^ ( 1 / 3 ) + ( 19-3 * sqrt ( 33 )) ^ ( 1 / 3 )) A ਫਰਮਾ:OEIS2C [1;1,5,4,2,305,1,8,2,1,4,6,14,3,1,13,5,1,7,...] 1.83928675521416113255185256465328660 0.36651 29205 81664 32701 Median of the Gumbel distribution l l 2 {\displaystyle {ll_{2}}} − ln ( ln ( 2 ) ) {\displaystyle -\ln(\ln(2))} A074785 [0;2,1,2,1,2,6,1,6,6,2,2,2,1,12,1,8,1,1,3,1,...] 0.36651292058166432701243915823266947 36.46215 96072 07911 7709 Pi^pi π π {\displaystyle \pi ^{\pi }} π π {\displaystyle \pi ^{\pi }} ਫਰਮਾ:OEIS2C [36;2,6,9,2,1,2,5,1,1,6,2,1,291,1,38,50,1,2,...] 36.4621596072079117709908260226921236 0.53964 54911 90413 18711 Ioachimescu constant 2 + ζ ( 1 2 ) {\displaystyle 2+\zeta ({\tfrac {1}{2}})} 2 − ( 1 + 2 ) ∑ n = 1 ∞ ( − 1 ) n + 1 n = γ + ∑ n = 1 ∞ ( − 1 ) 2 n γ n 2 n n ! {\displaystyle {2{-}(1{+}{\sqrt {2}})\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{\sqrt {n}}}}=\gamma +\sum _{n=1}^{\infty }{\frac {(-1)^{2n}\;\gamma _{n}}{2^{n}n!}}} γ + N [ sum [ n = 1 to ∞ ] {(( -1 ) ^ ( 2 n ) gamma_n ) / ( 2 ^ n n ! )}] 2- ਫਰਮਾ:OEIS2C [0;1,1,5,1,4,6,1,1,2,6,1,1,2,1,1,1,37,3,2,1,...] 0.53964549119041318711050084748470198 15.15426 22414 79264 1897 Exponential reiterated constant e e {\displaystyle e^{e}} ∑ n = 0 ∞ e n n ! = lim n → ∞ ( 1 + n n ) n − n ( 1 + n ) 1 + n {\displaystyle \sum _{n=0}^{\infty }{\frac {e^{n}}{n!}}=\lim _{n\to \infty }\left({\frac {1+n}{n}}\right)^{n^{-n}(1+n)^{1+n}}} ਫਰਮਾ:OEIS2C [15;6,2,13,1,3,6,2,1,1,5,1,1,1,9,4,1,1,1,6,7,...] 15.1542622414792641897604302726299119 0.64624 54398 94813 30426 Masser–Gramain constant C {\displaystyle {C}} γ β ( 1 ) + β ′ ( 1 ) = π ( − ln Γ ( 1 4 ) + 3 4 π + 1 2 ln 2 + 1 2 γ ) {\displaystyle \gamma {\beta }(1)\!+\!{\beta }'(1)\!=\pi \!\left(-\!\ln \Gamma ({\tfrac {1}{4}})+{\tfrac {3}{4}}\pi +{\tfrac {1}{2}}\ln 2+{\tfrac {1}{2}}\gamma \right)} = π ( − ln ( 1 4 ! ) + 3 4 ln π − 3 2 ln 2 + 1 2 γ ) {\displaystyle =\pi \!\left(-\!\ln({\tfrac {1}{4}}!)+{\tfrac {3}{4}}\ln \pi -{\tfrac {3}{2}}\ln 2+{\tfrac {1}{2}}\,\gamma \right)} γ = Euler–Mascheroni constant = 0.5772156649 … {\displaystyle \scriptstyle \gamma ={\text{Euler–Mascheroni constant}}=0.5772156649\ldots } β ( ) = Beta function , Γ ( ) = Gamma function {\displaystyle \scriptstyle \beta ()={\text{Beta function}},\quad \scriptstyle \Gamma ()={\text{Gamma function}}}
Pi / 4 * ( 2 * Gamma + 2 * Log [ 2 ] + 3 * Log [ Pi ] - 4 Log [ Gamma [ 1 / 4 ]]) ਫਰਮਾ:OEIS2C [0;1,1,1,4,1,3,2,3,9,1,33,1,4,3,3,5,3,1,3,4,...] 0.64624543989481330426647339684579279 1.11072 07345 39591 56175 The ratio of a square and circle circumscribed π 2 2 {\displaystyle {\frac {\pi }{2{\sqrt {2}}}}} ∑ n = 1 ∞ ( − 1 ) ⌊ n − 1 2 ⌋ 2 n + 1 = 1 1 + 1 3 − 1 5 − 1 7 + 1 9 + 1 11 − ⋯ {\displaystyle \sum _{n=1}^{\infty }{\frac {({-}1)^{\lfloor {\frac {n-1}{2}}\rfloor }}{2n+1}}={\frac {1}{1}}+{\frac {1}{3}}-{\frac {1}{5}}-{\frac {1}{7}}+{\frac {1}{9}}+{\frac {1}{11}}-{\cdots }} sum [ n = 1 to ∞ ] {( -1 ) ^ ( floor ( ( n -1 ) / 2 )) / ( 2 n -1 )} T ਫਰਮਾ:OEIS2C [1;9,31,1,1,17,2,3,3,2,3,1,1,2,2,1,4,9,1,3,...] 1.11072073453959156175397024751517342 1.45607 49485 82689 67139 Backhouse's constant B {\displaystyle {B}} lim k → ∞ | q k + 1 q k | where: Q ( x ) = 1 P ( x ) = ∑ k = 1 ∞ q k x k {\displaystyle \lim _{k\to \infty }\left|{\frac {q_{k+1}}{q_{k}}}\right\vert \quad \scriptstyle {\text{where:}}\displaystyle \;\;Q(x)={\frac {1}{P(x)}}=\!\sum _{k=1}^{\infty }q_{k}x^{k}} P ( x ) = ∑ k = 1 ∞ p k x k p k prime = 1 + 2 x + 3 x 2 + 5 x 3 + ⋯ {\displaystyle P(x)=\sum _{k=1}^{\infty }{\underset {p_{k}{\text{ prime}}}{p_{k}x^{k}}}=1+2x+3x^{2}+5x^{3}+\cdots }
1 / ( FindRoot [ 0 == 1 + Sum [ x ^ n Prime [ n ], { n , 10000 }], { x , { 1 }}) ਫਰਮਾ:OEIS2C [1;2,5,5,4,1,1,18,1,1,1,1,1,2,13,3,1,2,4,16,...] 1995 1.45607494858268967139959535111654355 1.85193 70519 82466 17036 Gibbs constant S i ( π ) {\displaystyle {Si(\pi )}} Sin integral ∫ 0 π sin t t d t = ∑ n = 1 ∞ ( − 1 ) n − 1 π 2 n − 1 ( 2 n − 1 ) ( 2 n − 1 ) ! {\displaystyle \int _{0}^{\pi }{\frac {\sin t}{t}}\,dt=\sum \limits _{n=1}^{\infty }(-1)^{n-1}{\frac {\pi ^{2n-1}}{(2n-1)(2n-1)!}}} = π − π 3 3 ⋅ 3 ! + π 5 5 ⋅ 5 ! − π 7 7 ⋅ 7 ! + ⋯ {\displaystyle =\pi -{\frac {\pi ^{3}}{3\cdot 3!}}+{\frac {\pi ^{5}}{5\cdot 5!}}-{\frac {\pi ^{7}}{7\cdot 7!}}+\cdots }
ਫਰਮਾ:OEIS2C [1;1,5,1,3,15,1,5,3,2,7,2,1,62,1,3,110,1,39,...] 1.85193705198246617036105337015799136 0.23571 11317 19232 93137 Copeland–Erdős constant C C E {\displaystyle {{\mathcal {C}}_{CE}}} ∑ n = 1 ∞ p n 10 n + ∑ k = 1 n ⌊ log 10 p k ⌋ {\displaystyle \sum _{n=1}^{\infty }{\frac {p_{n}}{10^{n+\sum \limits _{k=1}^{n}\lfloor \log _{10}{p_{k}}\rfloor }}}} sum [ n = 1 to ∞ ] { prime ( n ) / ( n + ( 10 ^ sum [ k = 1 to n ]{ floor ( log_10 prime ( k ))}))} I ਫਰਮਾ:OEIS2C [0;4,4,8,16,18,5,1,1,1,1,7,1,1,6,2,9,58,1,3,...] 0.23571113171923293137414347535961677 1.52362 70862 02492 10627 Fractal dimension of the boundary of the dragon curve C d {\displaystyle {C_{d}}} log ( 1 + 73 − 6 87 3 + 73 + 6 87 3 3 ) log ( 2 ) {\displaystyle {\frac {\log \left({\frac {1+{\sqrt[{3}]{73-6{\sqrt {87}}}}+{\sqrt[{3}]{73+6{\sqrt {87}}}}}{3}}\right)}{\log(2)}}} ( log (( 1 + ( 73-6 sqrt ( 87 )) ^ 1 / 3 + ( 73 + 6 sqrt ( 87 )) ^ 1 / 3 ) / 3 )) / log ( 2 ))) T [1;1,1,10,12,2,1,149,1,1,1,3,11,1,3,17,4,1,...] 1.52362708620249210627768393595421662 1.78221 39781 91369 11177 Grothendieck constant K R {\displaystyle {K_{R}}} π 2 log ( 1 + 2 ) {\displaystyle {\frac {\pi }{2\log(1+{\sqrt {2}})}}} ਫਰਮਾ:OEIS2C [1;1,3,1,1,2,4,2,1,1,17,1,12,4,3,5,10,1,1,3,...] 1.78221397819136911177441345297254934 1.58496 25007 21156 18145 Hausdorff dimension, Sierpinski triangle l o g 2 3 {\displaystyle {log_{2}3}} log 3 log 2 = ∑ n = 0 ∞ 1 2 2 n + 1 ( 2 n + 1 ) ∑ n = 0 ∞ 1 3 2 n + 1 ( 2 n + 1 ) = 1 2 + 1 24 + 1 160 + ⋯ 1 3 + 1 81 + 1 1215 + ⋯ {\displaystyle {\frac {\log 3}{\log 2}}={\frac {\sum _{n=0}^{\infty }{\frac {1}{2^{2n+1}(2n+1)}}}{\sum _{n=0}^{\infty }{\frac {1}{3^{2n+1}(2n+1)}}}}={\frac {{\frac {1}{2}}+{\frac {1}{24}}+{\frac {1}{160}}+\cdots }{{\frac {1}{3}}+{\frac {1}{81}}+{\frac {1}{1215}}+\cdots }}} ( Sum [ n = 0 to ∞ ] { 1 / ( 2 ^ ( 2 n + 1 ) ( 2 n + 1 ))}) / ( Sum [ n = 0 to ∞ ] { 1 / ( 3 ^ ( 2 n + 1 ) ( 2 n + 1 ))}) T ਫਰਮਾ:OEIS2C [1;1,1,2,2,3,1,5,2,23,2,2,1,1,55,1,4,3,1,1,...] 1.58496250072115618145373894394781651 1.30637 78838 63080 69046 Mills' constant θ {\displaystyle {\theta }} ⌊ θ 3 n ⌋ {\displaystyle \lfloor \theta ^{3^{n}}\rfloor } primes Nest [ NextPrime [ # ^ 3 ] & , 2 , 7 ] ^ ( 1 / 3 ^ 8 ) ਫਰਮਾ:OEIS2C [1;3,3,1,3,1,2,1,2,1,4,2,35,21,1,4,4,1,1,3,2,...] 1947 1.30637788386308069046861449260260571 2.02988 32128 19307 25004 Figure eight knot hyperbolic volume V 8 {\displaystyle {V_{8}}} 2 3 ∑ n = 1 ∞ 1 n ( 2 n n ) ∑ k = n 2 n − 1 1 k = 6 ∫ 0 π / 3 log ( 1 2 sin t ) d t = {\displaystyle 2{\sqrt {3}}\,\sum _{n=1}^{\infty }{\frac {1}{n{2n \choose n}}}\sum _{k=n}^{2n-1}{\frac {1}{k}}=6\int \limits _{0}^{\pi /3}\log \left({\frac {1}{2\sin t}}\right)\,dt=} 3 9 ∑ n = 0 ∞ ( − 1 ) n 27 n { 18 ( 6 n + 1 ) 2 − 18 ( 6 n + 2 ) 2 − 24 ( 6 n + 3 ) 2 − 6 ( 6 n + 4 ) 2 + 2 ( 6 n + 5 ) 2 } {\displaystyle \scriptstyle {\frac {\sqrt {3}}{9}}\,\sum \limits _{n=0}^{\infty }{\frac {(-1)^{n}}{27^{n}}}\,\left\{\!{\frac {18}{(6n+1)^{2}}}-{\frac {18}{(6n+2)^{2}}}-{\frac {24}{(6n+3)^{2}}}-{\frac {6}{(6n+4)^{2}}}+{\frac {2}{(6n+5)^{2}}}\!\right\}}
6 integral [ 0 to pi / 3 ] { log ( 1 / ( 2 sin ( n )))} ਫਰਮਾ:OEIS2C [2;33,2,6,2,1,2,2,5,1,1,7,1,1,1,113,1,4,5,1,...] 2.02988321281930725004240510854904057 262 53741 26407 68743 .99999 99999 99250 073 Hermite–Ramanujan constant R {\displaystyle {R}} e π 163 {\displaystyle e^{\pi {\sqrt {163}}}} T ਫਰਮਾ:OEIS2C [262537412640768743;1,1333462407511,1,8,1,1,5,...] 1859 262537412640768743.999999999999250073 1.74540 56624 07346 86349 Khinchin harmonic mean K − 1 {\displaystyle {K_{-1}}} log 2 ∑ n = 1 ∞ 1 n log ( 1 + 1 n ( n + 2 ) ) = lim n → ∞ n 1 a 1 + 1 a 2 + ⋯ + 1 a n {\displaystyle {\frac {\log 2}{\sum \limits _{n=1}^{\infty }{\frac {1}{n}}\log {\bigl (}1{+}{\frac {1}{n(n+2)}}{\bigr )}}}=\lim _{n\to \infty }{\frac {n}{{\frac {1}{a_{1}}}+{\frac {1}{a_{2}}}+\cdots +{\frac {1}{a_{n}}}}}} a 1 ... a n are elements of a continued fraction [a 0 ; a 1 , a 2 , ..., a n ]
( log 2 ) / ( sum [ n = 1 to ∞ ] { 1 / n log ( 1 + 1 / ( n ( n + 2 ))} ਫਰਮਾ:OEIS2C [1;1,2,1,12,1,5,1,5,13,2,13,2,1,9,1,6,1,3,1,...] 1.74540566240734686349459630968366106 1.64872 12707 00128 14684 Square root of the number e e {\displaystyle {\sqrt {e}}} ∑ n = 0 ∞ 1 2 n n ! = ∑ n = 0 ∞ 1 ( 2 n ) ! ! = 1 1 + 1 2 + 1 8 + 1 48 + ⋯ {\displaystyle \sum _{n=0}^{\infty }{\frac {1}{2^{n}n!}}=\sum _{n=0}^{\infty }{\frac {1}{(2n)!!}}={\frac {1}{1}}+{\frac {1}{2}}+{\frac {1}{8}}+{\frac {1}{48}}+\cdots } Sum [ n = 0 to ∞ ] { 1 / ( 2 ^ n n ! )} T ਫਰਮਾ:OEIS2C [1;1,1,1,5,1,1,9,1,1,13,1,1,17,1,1,21,1,1,...] = [1;1,1,1,4p+1 ], p∈ℕ 1.64872127070012814684865078781416357 1.01734 30619 84449 13971 Zeta(6) ζ ( 6 ) {\displaystyle \zeta (6)} π 6 945 = ∏ n = 1 ∞ 1 1 − p n − 6 p n : prime = 1 1 − 2 − 6 ⋅ 1 1 − 3 − 6 ⋅ 1 1 − 5 − 6 ⋯ {\displaystyle {\frac {\pi ^{6}}{945}}\!=\!\prod _{n=1}^{\infty }\!{\underset {p_{n}:{\text{ prime}}}{\frac {1}{{1-p_{n}}^{-6}}}}={\frac {1}{1\!-\!2^{-6}}}\!\cdot \!{\frac {1}{1\!-\!3^{-6}}}\!\cdot \!{\frac {1}{1\!-\!5^{-6}}}\cdots } Prod [ n = 1 to ∞ ] { 1 / ( 1 - ithprime ( n ) ^ -6 )} T ਫਰਮਾ:OEIS2C [1;57,1,1,1,15,1,6,3,61,1,5,3,1,6,1,3,3,6,1,...] 1.01734306198444913971451792979092052 0.10841 01512 23111 36151 Trott constant T 1 {\displaystyle \mathrm {T} _{1}} [ 1 , 0 , 8 , 4 , 1 , 0 , 1 , 5 , 1 , 2 , 2 , 3 , 1 , 1 , 1 , 3 , 6 , . . . ] {\displaystyle \textstyle [1,0,8,4,1,0,1,5,1,2,2,3,1,1,1,3,6,...]} 1 1 + 1 0 + 1 8 + 1 4 + 1 1 + 1 0 + 1 / ⋯ {\displaystyle {\tfrac {1}{1+{\tfrac {1}{0+{\tfrac {1}{8+{\tfrac {1}{4+{\tfrac {1}{1+{\tfrac {1}{0+1{/\cdots }}}}}}}}}}}}}}
ਫਰਮਾ:OEIS2C [0;9,4,2,5,1,2,2,3,1,1,1,3,6,1,5,1,1,2,...] 0.10841015122311136151129081140641509 0.00787 49969 97812 3844 Chaitin constant Ω {\displaystyle \Omega } ∑ p ∈ P 2 − | p | {\displaystyle \sum _{p\in P}2^{-|p|}} p : Halted program | p | : Size in bits of program p P : Domain of all programs that stop. T ਫਰਮਾ:OEIS2C [0; 126, 1, 62, 5, 5, 3, 3, 21, 1, 4, 1] 1975 0.0078749969978123844 0.83462 68416 74073 18628 Gauss constant G {\displaystyle {G}} 1 a g m ( 1 , 2 ) = 4 2 ( 1 4 ! ) 2 π 3 / 2 = 2 π ∫ 0 1 d x 1 − x 4 {\displaystyle {\frac {1}{\mathrm {agm} (1,{\sqrt {2}})}}={\frac {4{\sqrt {2}}\,({\tfrac {1}{4}}!)^{2}}{\pi ^{3/2}}}={\frac {2}{\pi }}\int _{0}^{1}{\frac {dx}{\sqrt {1-x^{4}}}}} AGM = Arithmetic–geometric mean
( 4 sqrt ( 2 )(( 1 / 4 ) ! ) ^ 2 ) / pi ^ ( 3 / 2 ) T ਫਰਮਾ:OEIS2C [0;1,5,21,3,4,14,1,1,1,1,1,3,1,15,1,3,7,1,...] 0.83462684167407318628142973279904680 1.45136 92348 83381 05028 Ramanujan–Soldner constant μ {\displaystyle {\mu }} l i ( x ) = ∫ 0 x d t ln t = 0 . . . . . . {\displaystyle \mathrm {li} (x)=\int \limits _{0}^{x}{\frac {dt}{\ln t}}=0{\color {White}{......}}} li = Logarithmic integral l i ( x ) = E i ( ln x ) . . . . . . . . {\displaystyle \mathrm {li} (x)\;=\;\mathrm {Ei} (\ln {x}){\color {White}{........}}} Ei = Exponential integral
I ਫਰਮਾ:OEIS2C [1;2,4,1,1,1,3,1,1,1,2,47,2,4,1,12,1,1,2,2,1,...] 1792 to 1809 1.45136923488338105028396848589202744 0.64341 05462 88338 02618 Cahen's constant ξ 2 {\displaystyle \xi _{2}} ∑ k = 1 ∞ ( − 1 ) k s k − 1 = 1 1 − 1 2 + 1 6 − 1 42 + 1 1806 ± ⋯ {\displaystyle \sum _{k=1}^{\infty }{\frac {(-1)^{k}}{s_{k}-1}}={\frac {1}{1}}-{\frac {1}{2}}+{\frac {1}{6}}-{\frac {1}{42}}+{\frac {1}{1806}}{\,\pm \cdots }} Where sk is the k th term of Sylvester's sequence 2, 3, 7, 43, 1807, ... Defined as: S 0 = 2 , S k = 1 + ∏ n = 0 k − 1 S n for k > 0 {\displaystyle \,\,S_{0}=\,2,\,\,S_{k}=\,1+\prod \limits _{n=0}^{k-1}S_{n}{\text{ for}}\;k>0}
T ਫਰਮਾ:OEIS2C [0; 1, 1, 1, 4, 9, 196, 16641, 639988804, ...] 1891 0.64341054628833802618225430775756476 1.41421 35623 73095 04880 Square root of 2, Pythagoras constant. 2 {\displaystyle {\sqrt {2}}} ∏ n = 1 ∞ ( 1 + ( − 1 ) n + 1 2 n − 1 ) = ( 1 + 1 1 ) ( 1 − 1 3 ) ( 1 + 1 5 ) ⋯ {\displaystyle \!\prod _{n=1}^{\infty }\!\left(1\!+\!{\frac {(-1)^{n+1}}{2n-1}}\right)\!=\!\left(1\!+\!{\frac {1}{1}}\right)\!\left(1\!-\!{\frac {1}{3}}\right)\!\left(1\!+\!{\frac {1}{5}}\right)\cdots } prod [ n = 1 to ∞ ] { 1 + ( -1 ) ^ ( n + 1 ) / ( 2 n -1 )} A ਫਰਮਾ:OEIS2C [1;2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,...] = [1;2 ...] 1.41421356237309504880168872420969808 1.77245 38509 05516 02729 Carlson–Levin constant Γ ( 1 2 ) {\displaystyle {\Gamma }({\tfrac {1}{2}})} π = ( − 1 2 ) ! = ∫ − ∞ ∞ 1 e x 2 d x = ∫ 0 1 1 − ln x d x {\displaystyle {\sqrt {\pi }}=\left(-{\frac {1}{2}}\right)!=\int _{-\infty }^{\infty }{\frac {1}{e^{x^{2}}}}\,dx=\int _{0}^{1}{\frac {1}{\sqrt {-\ln x}}}\,dx} T ਫਰਮਾ:OEIS2C [1;1,3,2,1,1,6,1,28,13,1,1,2,18,1,1,1,83,1,...] 1.77245385090551602729816748334114518 1.05946 30943 59295 26456 Musical interval between each half tone
2 12 {\displaystyle {\sqrt[{12}]{2}}} 2 x 12 0 1 2 3 4 5 6 7 8 9 10 11 12 Key C 1 C # D D # E F F # G G # A A # B C 2 {\displaystyle {\begin{array}{l|ccccccccccccr}\!2^{\frac {x}{12}}\!&\!\!\scriptstyle {0}&\!\!\!\!\scriptstyle {1}&\!\!\scriptstyle {2}&\!\!\scriptstyle {3}&\!\!\scriptstyle {4}&\scriptstyle {5}&\!\!\scriptstyle {6}&\!\!\scriptstyle {7}&\!\!\scriptstyle {8}&\!\!\scriptstyle {9}&\!\!\scriptstyle {10}&\!\!\scriptstyle {11}&\!\!\scriptstyle {12}\\\hline \!\scriptstyle {\textrm {Key}}\!&\!\scriptstyle {\mathrm {C_{1}} }&\!\!\scriptstyle {\mathrm {C^{\#}} }&\!\!\scriptstyle {\mathrm {D} }&\!\scriptstyle {\mathrm {D^{\#}} }&\!\!\scriptstyle {\mathrm {E} }&\scriptstyle {\mathrm {F} }&\!\scriptstyle {\mathrm {F^{\#}} }&\!\!\scriptstyle {\mathrm {G} }&\!\scriptstyle {\mathrm {G^{\#}} }&\!\!\scriptstyle {\mathrm {A} }&\!\scriptstyle {\mathrm {A^{\#}} }&\!\!\scriptstyle {\mathrm {B} }&\!\scriptstyle {\mathrm {C_{2}} }\end{array}}} (A = 440 Hz) A ਫਰਮਾ:OEIS2C [1;16,1,4,2,7,1,1,2,2,7,4,1,2,1,60,1,3,1,2,...] 1.05946309435929526456182529494634170 1.01494 16064 09653 62502 Gieseking constant π ln β {\displaystyle {\pi \ln \beta }} 3 3 4 ( 1 − ∑ n = 0 ∞ 1 ( 3 n + 2 ) 2 + ∑ n = 1 ∞ 1 ( 3 n + 1 ) 2 ) = {\displaystyle {\frac {3{\sqrt {3}}}{4}}\left(1-\sum _{n=0}^{\infty }{\frac {1}{(3n+2)^{2}}}+\sum _{n=1}^{\infty }{\frac {1}{(3n+1)^{2}}}\right)=} 3 3 4 ( 1 − 1 2 2 + 1 4 2 − 1 5 2 + 1 7 2 − 1 8 2 + 1 10 2 ± ⋯ ) {\displaystyle \textstyle {\frac {3{\sqrt {3}}}{4}}\left(1-{\frac {1}{2^{2}}}+{\frac {1}{4^{2}}}-{\frac {1}{5^{2}}}+{\frac {1}{7^{2}}}-{\frac {1}{8^{2}}}+{\frac {1}{10^{2}}}\pm \cdots \right)} .
sqrt ( 3 ) * 3 / 4 * ( 1 - Sum [ n = 0 to ∞ ] { 1 / (( 3 n + 2 ) ^ 2 )} + Sum [ n = 1 to ∞ ] { 1 / (( 3 n + 1 ) ^ 2 )}) ਫਰਮਾ:OEIS2C [1;66,1,12,1,2,1,4,2,1,3,3,1,4,1,56,2,2,11,...] 1912 1.01494160640965362502120255427452028 2.62205 75542 92119 81046 Lemniscate constant ϖ {\displaystyle {\varpi }} π G = 4 2 π Γ ( 5 4 ) 2 = 1 4 2 π Γ ( 1 4 ) 2 = 4 2 π ( 1 4 ! ) 2 {\displaystyle \pi \,{G}=4{\sqrt {\tfrac {2}{\pi }}}\,\Gamma {\left({\tfrac {5}{4}}\right)^{2}}={\tfrac {1}{4}}{\sqrt {\tfrac {2}{\pi }}}\,\Gamma {\left({\tfrac {1}{4}}\right)^{2}}=4{\sqrt {\tfrac {2}{\pi }}}\left({\tfrac {1}{4}}!\right)^{2}} T ਫਰਮਾ:OEIS2C [2;1,1,1,1,1,4,1,2,5,1,1,1,14,9,2,6,2,9,4,1,...] 1798 2.62205755429211981046483958989111941 1.28242 71291 00622 63687 Glaisher–Kinkelin constant A {\displaystyle {A}} e 1 12 − ζ ′ ( − 1 ) = e 1 8 − 1 2 ∑ n = 0 ∞ 1 n + 1 ∑ k = 0 n ( − 1 ) k ( n k ) ( k + 1 ) 2 ln ( k + 1 ) {\displaystyle e^{{\frac {1}{12}}-\zeta ^{\prime }(-1)}=e^{{\frac {1}{8}}-{\frac {1}{2}}\sum \limits _{n=0}^{\infty }{\frac {1}{n+1}}\sum \limits _{k=0}^{n}\left(-1\right)^{k}{\binom {n}{k}}\left(k+1\right)^{2}\ln(k+1)}} T ? ਫਰਮਾ:OEIS2C [1;3,1,1,5,1,1,1,3,12,4,1,271,1,1,2,7,1,35,...] 1.28242712910062263687534256886979172 -4.22745 35333 76265 408 Digamma (1/4) ψ ( 1 4 ) {\displaystyle {\psi }({\tfrac {1}{4}})} − γ − π 2 − 3 ln 2 = − γ + ∑ n = 0 ∞ ( 1 n + 1 − 1 n + 1 4 ) {\displaystyle -\gamma -{\frac {\pi }{2}}-3\ln {2}=-\gamma +\sum _{n=0}^{\infty }\left({\frac {1}{n+1}}-{\frac {1}{n+{\tfrac {1}{4}}}}\right)} - EulerGamma - \ pi / 2 -3 log 2 ਫਰਮਾ:OEIS2C -[4;4,2,1,1,10,1,5,9,11,1,22,1,1,14,1,2,1,4,...] -4.2274535333762654080895301460966835 0.28674 74284 34478 73410 Strongly Carefree constant K 2 {\displaystyle K_{2}} ∏ n = 1 ∞ ( 1 − 3 p n − 2 p n 3 ) p n : prime = 6 π 2 ∏ n = 1 ∞ ( 1 − 1 p n ( p n + 1 ) ) p n : prime {\displaystyle \prod _{n=1}^{\infty }{\underset {p_{n}:{\text{ prime}}}{\left(1-{\frac {3p_{n}-2}{{p_{n}}^{3}}}\right)}}={\frac {6}{\pi ^{2}}}\prod _{n=1}^{\infty }{\underset {p_{n}:{\text{ prime}}}{\left(1-{\frac {1}{p_{n}(p_{n}+1)}}\right)}}} N [ prod [ k = 1 to ∞ ] { 1 - ( 3 * prime ( k ) -2 ) / ( prime ( k ) ^ 3 )}] ਫਰਮਾ:OEIS2C [0;3,2,19,3,12,1,5,1,5,1,5,2,1,1,1,1,1,3,7,...] 0.28674742843447873410789271278983845 3.62560 99082 21908 31193 Gamma(1/4) Γ ( 1 4 ) {\displaystyle \Gamma ({\tfrac {1}{4}})} 4 ( 1 4 ) ! = ( − 3 4 ) ! {\displaystyle 4\left({\frac {1}{4}}\right)!=\left(-{\frac {3}{4}}\right)!} T ਫਰਮਾ:OEIS2C [3;1,1,1,2,25,4,9,1,1,8,4,1,6,1,1,19,1,1,4,1,...] 1729 3.62560990822190831193068515586767200 1.66168 79496 33594 12129 Somos' quadratic recurrence constant σ {\displaystyle {\sigma }} ∏ n = 1 ∞ n 1 / 2 n = 1 2 3 ⋯ = 1 1 / 2 2 1 / 4 3 1 / 8 ⋯ {\displaystyle \prod _{n=1}^{\infty }n^{{1/2}^{n}}={\sqrt {1{\sqrt {2{\sqrt {3\cdots }}}}}}=1^{1/2}\;2^{1/4}\;3^{1/8}\cdots } prod [ n = 1 to ∞ ] { n ^ ( 1 / 2 ) ^ n } T ? ਫਰਮਾ:OEIS2C [1;1,1,1,21,1,1,1,6,4,2,1,1,2,1,3,1,13,13,...] 1.66168794963359412129581892274995074 0.95531 66181 245092 78163 Magic angle θ m {\displaystyle {\theta _{m}}} arctan ( 2 ) = arccos ( 1 3 ) ≈ 54.7356 ∘ {\displaystyle \arctan \left({\sqrt {2}}\right)=\arccos \left({\sqrt {\tfrac {1}{3}}}\right)\approx \textstyle {54.7356}^{\circ }} T ਫਰਮਾ:OEIS2C [0;1,21,2,1,1,1,2,1,2,2,4,1,2,9,1,2,1,1,1,3,...] 0.95531661812450927816385710251575775 1.78107 24179 90197 98523 Exp.gamma, Barnes G-function e γ {\displaystyle e^{\gamma }} ∏ n = 1 ∞ e 1 n 1 + 1 n = ∏ n = 0 ∞ ( ∏ k = 0 n ( k + 1 ) ( − 1 ) k + 1 ( n k ) ) 1 n + 1 = {\displaystyle \prod _{n=1}^{\infty }{\frac {e^{\frac {1}{n}}}{1+{\tfrac {1}{n}}}}=\prod _{n=0}^{\infty }\left(\prod _{k=0}^{n}(k+1)^{(-1)^{k+1}{n \choose k}}\right)^{\frac {1}{n+1}}=} ( 2 1 ) 1 / 2 ( 2 2 1 ⋅ 3 ) 1 / 3 ( 2 3 ⋅ 4 1 ⋅ 3 3 ) 1 / 4 ( 2 4 ⋅ 4 4 1 ⋅ 3 6 ⋅ 5 ) 1 / 5 ⋯ {\displaystyle \textstyle \left({\frac {2}{1}}\right)^{1/2}\left({\frac {2^{2}}{1\cdot 3}}\right)^{1/3}\left({\frac {2^{3}\cdot 4}{1\cdot 3^{3}}}\right)^{1/4}\left({\frac {2^{4}\cdot 4^{4}}{1\cdot 3^{6}\cdot 5}}\right)^{1/5}\cdots }
Prod [ n = 1 to ∞ ] { e ^ ( 1 / n )} / { 1 + 1 / n } ਫਰਮਾ:OEIS2C [1;1,3,1,1,3,5,4,1,1,2,2,1,7,9,1,16,1,1,1,2,...] 1.78107241799019798523650410310717954 0.74759 79202 53411 43517 Rényi's Parking Constant m {\displaystyle {m}} ∫ 0 ∞ e x p ( − 2 ∫ 0 x 1 − e − y y d y ) d x = e − 2 γ ∫ 0 ∞ e − 2 Γ ( 0 , n ) n 2 {\displaystyle \int \limits _{0}^{\infty }exp\left(\!-2\int \limits _{0}^{x}{\frac {1-e^{-y}}{y}}dy\right)\!dx={e^{-2\gamma }}\int \limits _{0}^{\infty }{\frac {e^{-2\Gamma (0,n)}}{n^{2}}}} [ e ^ ( -2 * Gamma )] * । nt { n , 0 , ∞ }[ e ^ ( - 2 * Gamma ( 0 , n )) / n ^ 2 ] ਫਰਮਾ:OEIS2C [0;1,2,1,25,3,1,2,1,1,12,1,2,1,1,3,1,2,1,43,...] 0.74759792025341143517873094383017817 1.27323 95447 35162 68615 Ramanujan–Forsyth series 4 π {\displaystyle {\frac {4}{\pi }}} ∑ n = 0 ∞ ( ( 2 n − 3 ) ! ! ( 2 n ) ! ! ) 2 = 1 + ( 1 2 ) 2 + ( 1 2 ⋅ 4 ) 2 + ( 1 ⋅ 3 2 ⋅ 4 ⋅ 6 ) 2 + ⋯ {\displaystyle \displaystyle \sum \limits _{n=0}^{\infty }\textstyle \left({\frac {(2n-3)!!}{(2n)!!}}\right)^{2}={1\!+\!\left({\frac {1}{2}}\right)^{2}\!+\!\left({\frac {1}{2\cdot 4}}\right)^{2}\!+\!\left({\frac {1\cdot 3}{2\cdot 4\cdot 6}}\right)^{2}+\cdots }} Sum [ n = 0 to ∞ ] {[( 2 n -3 ) !! / ( 2 n ) !! ] ^ 2 } I ਫਰਮਾ:OEIS2C [1;3,1,1,1,15,2,72,1,9,1,17,1,2,1,5,1,1,10,...] 1.27323954473516268615107010698011489 1.44466 78610 09766 13365 Steiner number, Iterated exponential Constant e e {\displaystyle {\sqrt[{e}]{e}}} e 1 e . . . . . . . . . . . {\displaystyle e^{\frac {1}{e}}{\color {White}{...........}}} = Upper Limit of Tetration T ਫਰਮਾ:OEIS2C [1;2,4,55,27,1,1,16,9,3,2,8,3,2,1,1,4,1,9,...] 1.44466786100976613365833910859643022 0.69220 06275 55346 35386 Minimum value of función ƒ (x) = xx ( 1 e ) 1 e {\displaystyle {\left({\frac {1}{e}}\right)}^{\frac {1}{e}}} e − 1 e . . . . . . . . . . {\displaystyle {e}^{-{\frac {1}{e}}}{\color {White}{..........}}} =।nverse Steiner Number ਫਰਮਾ:OEIS2C [0;1,2,4,55,27,1,1,16,9,3,2,8,3,2,1,1,4,1,9,...] 0.69220062755534635386542199718278976 0.34053 73295 50999 14282 Pólya Random walk constant p ( 3 ) {\displaystyle {p(3)}} 1 − ( 3 ( 2 π ) 3 ∫ − π π ∫ − π π ∫ − π π d x d y d z 3 − cos x − cos y − cos z ) − 1 {\displaystyle 1-\!\!\left({3 \over (2\pi )^{3}}\int \limits _{-\pi }^{\pi }\int \limits _{-\pi }^{\pi }\int \limits _{-\pi }^{\pi }{dx\,dy\,dz \over 3-\!\cos x-\!\cos y-\!\cos z}\right)^{\!-1}} = 1 − 16 2 3 π 3 ( Γ ( 1 24 ) Γ ( 5 24 ) Γ ( 7 24 ) Γ ( 11 24 ) ) − 1 {\displaystyle =1-16{\sqrt {\tfrac {2}{3}}}\;\pi ^{3}\left(\Gamma ({\tfrac {1}{24}})\Gamma ({\tfrac {5}{24}})\Gamma ({\tfrac {7}{24}})\Gamma ({\tfrac {11}{24}})\right)^{-1}}
1-16 * Sqrt [ 2 / 3 ] * Pi ^ 3 / ( Gamma [ 1 / 24 ] * Gamma [ 5 / 24 ] * Gamma [ 7 / 24 ] * Gamma [ 11 / 24 ]) ਫਰਮਾ:OEIS2C [0;2,1,14,1,3,8,1,5,2,7,1,12,1,5,59,1,1,1,3,...] 0.34053732955099914282627318443290289 0.54325 89653 42976 70695 Bloch–Landau constant L {\displaystyle {L}} = Γ ( 1 3 ) Γ ( 5 6 ) Γ ( 1 6 ) = ( − 2 3 ) ! ( − 1 + 5 6 ) ! ( − 1 + 1 6 ) ! {\displaystyle ={\frac {\Gamma ({\tfrac {1}{3}})\;\Gamma ({\tfrac {5}{6}})}{\Gamma ({\tfrac {1}{6}})}}={\frac {(-{\tfrac {2}{3}})!\;(-1+{\tfrac {5}{6}})!}{(-1+{\tfrac {1}{6}})!}}} gamma ( 1 / 3 ) * gamma ( 5 / 6 ) / gamma ( 1 / 6 ) ਫਰਮਾ:OEIS2C [0;1,1,5,3,1,1,2,1,1,6,3,1,8,11,2,1,1,27,4,...] 1929 0.54325896534297670695272829530061323 0.18785 96424 62067 12024 MRB Constant, Marvin Ray Burns C M R B {\displaystyle C_{{}_{MRB}}} ∑ n = 1 ∞ ( − 1 ) n ( n 1 / n − 1 ) = − 1 1 + 2 2 − 3 3 + ⋯ {\displaystyle \sum _{n=1}^{\infty }(-1)^{n}(n^{1/n}-1)=-{\sqrt[{1}]{1}}+{\sqrt[{2}]{2}}-{\sqrt[{3}]{3}}+\cdots } Sum [ n = 1 to ∞ ] {( -1 ) ^ n ( n ^ ( 1 / n ) -1 )} ਫਰਮਾ:OEIS2C [0;5,3,10,1,1,4,1,1,1,1,9,1,1,12,2,17,2,2,1,...] 1999 0.18785964246206712024851793405427323 1.46707 80794 33975 47289 Porter Constant C {\displaystyle {C}} 6 ln 2 π 2 ( 3 ln 2 + 4 γ − 24 π 2 ζ ′ ( 2 ) − 2 ) − 1 2 {\displaystyle {\frac {6\ln 2}{\pi ^{2}}}\left(3\ln 2+4\,\gamma -{\frac {24}{\pi ^{2}}}\,\zeta '(2)-2\right)-{\frac {1}{2}}} γ = Euler–Mascheroni Constant = 0.5772156649 … {\displaystyle \scriptstyle \gamma \,{\text{= Euler–Mascheroni Constant}}=0.5772156649\ldots } ζ ′ ( 2 ) = Derivative of ζ ( 2 ) = − ∑ n = 2 ∞ ln n n 2 = − 0.9375482543 … {\displaystyle \scriptstyle \zeta '(2)\,{\text{= Derivative of }}\zeta (2)=-\sum \limits _{n=2}^{\infty }{\frac {\ln n}{n^{2}}}=-0.9375482543\ldots }
6 * ln2 / pi ^ 2 ( 3 * ln2 + 4 EulerGamma - WeierstrassZeta ' ( 2 ) * 24 / pi ^ 2-2 ) -1 / 2 ਫਰਮਾ:OEIS2C [1;2,7,10,1,2,38,5,4,1,4,12,5,1,5,1,2,3,1,...] 1974 1.46707807943397547289779848470722995 4.66920 16091 02990 67185 Feigenbaum constant δ δ {\displaystyle {\delta }} lim n → ∞ x n + 1 − x n x n + 2 − x n + 1 x ∈ ( 3.8284 ; 3.8495 ) {\displaystyle \lim _{n\to \infty }{\frac {x_{n+1}-x_{n}}{x_{n+2}-x_{n+1}}}\qquad \scriptstyle x\in (3.8284;\,3.8495)} x n + 1 = a x n ( 1 − x n ) or x n + 1 = a sin ( x n ) {\displaystyle \scriptstyle x_{n+1}=\,ax_{n}(1-x_{n})\quad {\text{or}}\quad x_{n+1}=\,a\sin(x_{n})}
T ਫਰਮਾ:OEIS2C [4;1,2,43,2,163,2,3,1,1,2,5,1,2,3,80,2,5,...] 1975 4.66920160910299067185320382046620161 2.50290 78750 95892 82228 Feigenbaum constant α α {\displaystyle \alpha } lim n → ∞ d n d n + 1 {\displaystyle \lim _{n\to \infty }{\frac {d_{n}}{d_{n+1}}}} T ? ਫਰਮਾ:OEIS2C [2;1,1,85,2,8,1,10,16,3,8,9,2,1,40,1,2,3,...] 1979 2.50290787509589282228390287321821578 0.62432 99885 43550 87099 Golomb–Dickman constant λ {\displaystyle {\lambda }} ∫ 0 ∞ f ( x ) x 2 d x Para x > 2 = ∫ 0 1 e Li ( n ) d n Li: Logarithmic integral {\displaystyle \int \limits _{0}^{\infty }{\underset {{\text{Para }}x>2}{{\frac {f(x)}{x^{2}}}\,dx}}=\int \limits _{0}^{1}e^{\operatorname {Li} (n)}dn\quad \scriptstyle {\text{Li: Logarithmic integral}}} N [ Int { n , 0 , 1 }[ e ^ Li ( n )], 34 ] ਫਰਮਾ:OEIS2C [0;1,1,1,1,1,22,1,2,3,1,1,11,1,1,2,22,2,6,1,...] 1930 & 1964 0.62432998854355087099293638310083724 23.14069 26327 79269 0057 Gelfond constant e π {\displaystyle {e}^{\pi }} ( − 1 ) − i = i − 2 i = ∑ n = 0 ∞ π n n ! = π 1 1 + π 2 2 ! + π 3 3 ! + ⋯ {\displaystyle (-1)^{-i}=i^{-2i}=\sum _{n=0}^{\infty }{\frac {\pi ^{n}}{n!}}={\frac {\pi ^{1}}{1}}+{\frac {\pi ^{2}}{2!}}+{\frac {\pi ^{3}}{3!}}+\cdots } Sum [ n = 0 to ∞ ] {( pi ^ n ) / n ! } T ਫਰਮਾ:OEIS2C [23;7,9,3,1,1,591,2,9,1,2,34,1,16,1,30,1,...] 23.1406926327792690057290863679485474 7.38905 60989 30650 22723 Conic constant, Schwarzschild constant e 2 {\displaystyle e^{2}} ∑ n = 0 ∞ 2 n n ! = 1 + 2 + 2 2 2 ! + 2 3 3 ! + 2 4 4 ! + 2 5 5 ! + ⋯ {\displaystyle \sum _{n=0}^{\infty }{\frac {2^{n}}{n!}}=1+2+{\frac {2^{2}}{2!}}+{\frac {2^{3}}{3!}}+{\frac {2^{4}}{4!}}+{\frac {2^{5}}{5!}}+\cdots } T ਫਰਮਾ:OEIS2C [7;2,1,1,3,18,5,1,1,6,30,8,1,1,9,42,11,1,...] = [7,2,1,1,n,4*n+6,n+2 ], n = 3, 6, 9, etc. 7.38905609893065022723042746057500781 0.35323 63718 54995 98454 Hafner–Sarnak–McCurley constant (1) σ {\displaystyle {\sigma }} ∏ k = 1 ∞ { 1 − [ 1 − ∏ j = 1 n ( 1 − p k − j ) ] 2 p k : prime } {\displaystyle \prod _{k=1}^{\infty }\left\{1-[1-\prod _{j=1}^{n}{\underset {p_{k}:{\text{ prime}}}{(1-p_{k}^{-j})]^{2}}}\right\}} prod [ k = 1 to ∞ ] { 1 - ( 1 - prod [ j = 1 to n ] { 1 - ithprime ( k ) ^- j }) ^ 2 } ਫਰਮਾ:OEIS2C [0;2,1,4,1,10,1,8,1,4,1,2,1,2,1,2,6,1,1,1,3,...] 1993 0.35323637185499598454351655043268201 0.60792 71018 54026 62866 Hafner–Sarnak–McCurley constant (2) 1 ζ ( 2 ) {\displaystyle {\frac {1}{\zeta (2)}}} 6 π 2 = ∏ n = 0 ∞ ( 1 − 1 p n 2 ) p n : prime = ( 1 − 1 2 2 ) ( 1 − 1 3 2 ) ( 1 − 1 5 2 ) ⋯ {\displaystyle {\frac {6}{\pi ^{2}}}=\prod _{n=0}^{\infty }{\underset {p_{n}:{\text{ prime}}}{\!\left(\!1-{\frac {1}{{p_{n}}^{2}}}\!\right)}}\!=\!\textstyle \left(1\!-\!{\frac {1}{2^{2}}}\right)\!\left(1\!-\!{\frac {1}{3^{2}}}\right)\!\left(1\!-\!{\frac {1}{5^{2}}}\right)\cdots } Prod { n = 1 to ∞ } ( 1-1 / ithprime ( n ) ^ 2 ) T ਫਰਮਾ:OEIS2C [0;1,1,1,1,4,2,4,7,1,4,2,3,4,10,1,2,1,1,1,...] 0.60792710185402662866327677925836583 0.12345 67891 01112 13141 Champernowne constant C 10 {\displaystyle C_{10}} ∑ n = 1 ∞ ∑ k = 10 n − 1 10 n − 1 k 10 k n − 9 ∑ j = 0 n − 1 10 j ( n − j − 1 ) {\displaystyle \sum _{n=1}^{\infty }\;\sum _{k=10^{n-1}}^{10^{n}-1}{\frac {k}{10^{kn-9\sum _{j=0}^{n-1}10^{j}(n-j-1)}}}} T ਫਰਮਾ:OEIS2C [0;8,9,1,149083,1,1,1,4,1,1,1,3,4,1,1,1,15,...] 1933 0.12345678910111213141516171819202123 0.76422 36535 89220 66299 Landau-Ramanujan constant K {\displaystyle K} 1 2 ∏ p ≡ 3 mod 4 ( 1 − 1 p 2 ) − 1 2 p : prime = π 4 ∏ p ≡ 1 mod 4 ( 1 − 1 p 2 ) 1 2 p : prime {\displaystyle {\frac {1}{\sqrt {2}}}\prod _{p\equiv 3\!\!\!\!\!\mod \!4}\!\!{\underset {\!\!\!\!\!\!\!\!p:{\text{ prime}}}{\left(1-{\frac {1}{p^{2}}}\right)^{-{\frac {1}{2}}}}}\!\!={\frac {\pi }{4}}\prod _{p\equiv 1\!\!\!\!\!\mod \!4}\!\!{\underset {\!\!\!\!p:{\text{ prime}}}{\left(1-{\frac {1}{p^{2}}}\right)^{\frac {1}{2}}}}} T ? ਫਰਮਾ:OEIS2C [0;1,3,4,6,1,15,1,2,2,3,1,23,3,1,1,3,1,1,6,4,...] 0.76422365358922066299069873125009232 2.71828 18284 59045 23536 Number e, Euler's number e {\displaystyle {e}} lim n → ∞ ( 1 + 1 n ) n = ∑ n = 0 ∞ 1 n ! = 1 0 ! + 1 1 + 1 2 ! + 1 3 ! + ⋯ {\displaystyle \!\lim _{n\to \infty }\!\left(\!1\!+\!{\frac {1}{n}}\right)^{n}\!=\!\sum _{n=0}^{\infty }{\frac {1}{n!}}={\frac {1}{0!}}+{\frac {1}{1}}+{\frac {1}{2!}}+{\frac {1}{3!}}+\textstyle \cdots } Sum [ n = 0 to ∞ ] { 1 / n ! } (* lim_(n->∞) (1+1/n)^n *) T ਫਰਮਾ:OEIS2C [2;1,2,1,1,4,1,1,6,1,1,8,1,1,10,1,1,12,1,...] = [2;1,2p,1 ], p∈ℕ 2.71828182845904523536028747135266250 0.36787 94411 71442 32159 Inverse of Number e 1 e {\displaystyle {\frac {1}{e}}} ∑ n = 0 ∞ ( − 1 ) n n ! = 1 0 ! − 1 1 ! + 1 2 ! − 1 3 ! + 1 4 ! − 1 5 ! + ⋯ {\displaystyle \sum _{n=0}^{\infty }{\frac {(-1)^{n}}{n!}}={\frac {1}{0!}}-{\frac {1}{1!}}+{\frac {1}{2!}}-{\frac {1}{3!}}+{\frac {1}{4!}}-{\frac {1}{5!}}+\cdots } Sum [ n = 2 to ∞ ] {( -1 ) ^ n / n ! } T ਫਰਮਾ:OEIS2C [0;2,1,1,2,1,1,4,1,1,6,1,1,8,1,1,10,1,1,12,...] = [0;2,1,1,2p,1 ], p∈ℕ 1618 0.36787944117144232159552377016146086 0.69034 71261 14964 31946 Upper iterated exponential H 2 n + 1 {\displaystyle {H}_{2n+1}} lim n → ∞ H 2 n + 1 = ( 1 2 ) ( 1 3 ) ( 1 4 ) ⋅ ⋅ ( 1 2 n + 1 ) = 2 − 3 − 4 ⋅ ⋅ − 2 n − 1 {\displaystyle \lim _{n\to \infty }{H}_{2n+1}=\textstyle \left({\frac {1}{2}}\right)^{\left({\frac {1}{3}}\right)^{\left({\frac {1}{4}}\right)^{\cdot ^{\cdot ^{\left({\frac {1}{2n+1}}\right)}}}}}={2}^{-3^{-4^{\cdot ^{\cdot ^{-2n-1}}}}}} 2 ^ -3 ^ -4 ^ -5 ^ -6 ^ -7 ^ -8 ^ -9 ^ -10 ^ -11 ^ -12 ^ -13 … ਫਰਮਾ:OEIS2C [0;1,2,4,2,1,3,1,2,2,1,4,1,2,4,3,1,1,10,1,3,2,...] 0.69034712611496431946732843846418942 0.65836 55992 ... Lower límit iterated exponential H 2 n {\displaystyle {H}_{2n}} lim n → ∞ H 2 n = ( 1 2 ) ( 1 3 ) ( 1 4 ) ⋅ ⋅ ( 1 2 n ) = 2 − 3 − 4 ⋅ ⋅ − 2 n {\displaystyle \lim _{n\to \infty }{H}_{2n}=\textstyle \left({\frac {1}{2}}\right)^{\left({\frac {1}{3}}\right)^{\left({\frac {1}{4}}\right)^{\cdot ^{\cdot ^{\left({\frac {1}{2n}}\right)}}}}}={2}^{-3^{-4^{\cdot ^{\cdot ^{-2n}}}}}} 2 ^ -3 ^ -4 ^ -5 ^ -6 ^ -7 ^ -8 ^ -9 ^ -10 ^ -11 ^ -12 … [0;1,1,1,12,1,2,1,1,4,3,1,1,2,1,2,1,51,2,2,1,...] 0.6583655992... 3.14159 26535 89793 23846 π number, Archimedes number π {\displaystyle \pi } lim n → ∞ 2 n 2 − 2 + 2 + ⋯ + 2 ⏟ n {\displaystyle \lim _{n\to \infty }\,2^{n}\underbrace {\sqrt {2-{\sqrt {2+{\sqrt {2+\cdots +{\sqrt {2}}}}}}}} _{n}} Sum [ n = 0 to ∞ ] {( -1 ) ^ n 4 / ( 2 n + 1 )} T ਫਰਮਾ:OEIS2C [3;7,15,1,292,1,1,1,2,1,3,1,14,2,1,1,2,2,2,...] 3.14159265358979323846264338327950288 1.92878 00... Wright constant ω {\displaystyle {\omega }} ⌊ 2 2 2 ⋅ ⋅ 2 ω ⌋ = primes: ⌊ 2 ω ⌋ =3, ⌊ 2 2 ω ⌋ =13, ⌊ 2 2 2 ω ⌋ = 16381 , … {\displaystyle \left\lfloor 2^{2^{2^{\cdot ^{\cdot ^{2^{\omega }}}}}}\!\right\rfloor \scriptstyle {\text{= primes:}}\displaystyle \left\lfloor 2^{\omega }\right\rfloor \scriptstyle {\text{=3,}}\displaystyle \left\lfloor 2^{2^{\omega }}\right\rfloor \scriptstyle {\text{=13,}}\displaystyle \left\lfloor 2^{2^{2^{\omega }}}\right\rfloor \scriptstyle =16381,\ldots } ਫਰਮਾ:OEIS2C [1; 1, 13, 24, 2, 1, 1, 3, 1, 1, 3] 1.9287800... 0.46364 76090 00806 11621 Machin–Gregory series arctan 1 2 {\displaystyle \arctan {\frac {1}{2}}} ∑ n = 0 ∞ ( − 1 ) n x 2 n + 1 2 n + 1 = 1 2 − 1 3 ⋅ 2 3 + 1 5 ⋅ 2 5 − 1 7 ⋅ 2 7 + ⋯ For x = 1 / 2 {\displaystyle {\underset {{\text{For }}x=1/2\qquad \qquad }{\sum _{n=0}^{\infty }{\frac {(\!-1\!)^{n}\,x^{2n+1}}{2n+1}}={\frac {1}{2}}{-}{\frac {1}{3\!\cdot \!2^{3}}}{+}{\frac {1}{5\!\cdot \!2^{5}}}{-}{\frac {1}{7\!\cdot \!2^{7}}}{+}\cdots }}} Sum [ n = 0 to ∞ ] {( -1 ) ^ n ( 1 / 2 ) ^ ( 2 n + 1 ) / ( 2 n + 1 )} I ਫਰਮਾ:OEIS2C [0;2,6,2,1,1,1,6,1,2,1,1,2,10,1,2,1,2,1,1,1,...] 0.46364760900080611621425623146121440 0.69777 46579 64007 98200 Continued fraction constant, Bessel function C C F {\displaystyle {C}_{CF}} I 1 ( 2 ) I 0 ( 2 ) = ∑ n = 0 ∞ n n ! n ! ∑ n = 0 ∞ 1 n ! n ! = 1 1 + 1 2 + 1 3 + 1 4 + 1 5 + 1 6 + 1 / ⋯ {\displaystyle {\frac {I_{1}(2)}{I_{0}(2)}}={\frac {\sum \limits _{n=0}^{\infty }{\frac {n}{n!n!}}}{\sum \limits _{n=0}^{\infty }{\frac {1}{n!n!}}}}=\textstyle {\tfrac {1}{1+{\tfrac {1}{2+{\tfrac {1}{3+{\tfrac {1}{4+{\tfrac {1}{5+{\tfrac {1}{6+1{/\cdots }}}}}}}}}}}}}} ( Sum [ n = 0 to ∞ ] { n / ( n ! n ! )}) / ( Sum [ n = 0 to ∞ ] { 1 / ( n ! n ! )}) I ਫਰਮਾ:OEIS2C [0;1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,...] = [0;p+1 ], p∈ℕ 0.69777465796400798200679059255175260 1.90216 05831 04 Brun 2 constant = Σ inverse of Twin primes B 2 {\displaystyle {B}_{\,2}} ∑ ( 1 p + 1 p + 2 ) p , p + 2 : prime = ( 1 3 + 1 5 ) + ( 1 5 + 1 7 ) + ( 1 11 + 1 13 ) + ⋯ {\displaystyle \textstyle {\underset {p,\,p+2:{\text{ prime}}}{\sum ({\frac {1}{p}}+{\frac {1}{p+2}})}}=({\frac {1}{3}}\!+\!{\frac {1}{5}})+({\tfrac {1}{5}}\!+\!{\tfrac {1}{7}})+({\tfrac {1}{11}}\!+\!{\tfrac {1}{13}})+\cdots } ਫਰਮਾ:OEIS2C [1; 1, 9, 4, 1, 1, 8, 3, 4, 4, 2, 2] 1.902160583104 0.87058 83799 75 Brun 4 constant = Σ inv.prime quadruplets B 4 {\displaystyle {B}_{\,4}} ∑ ( 1 p + 1 p + 2 + 1 p + 6 + 1 p + 8 ) p , p + 2 , p + 6 , p + 8 : prime {\displaystyle \textstyle {\sum ({\frac {1}{p}}+{\frac {1}{p+2}}+{\frac {1}{p+6}}+{\frac {1}{p+8}})}\scriptstyle \quad {p,\;p+2,\;p+6,\;p+8:{\text{ prime}}}} ( 1 5 + 1 7 + 1 11 + 1 13 ) + ( 1 11 + 1 13 + 1 17 + 1 19 ) + … {\displaystyle \textstyle {\left({\tfrac {1}{5}}+{\tfrac {1}{7}}+{\tfrac {1}{11}}+{\tfrac {1}{13}}\right)}+\left({\tfrac {1}{11}}+{\tfrac {1}{13}}+{\tfrac {1}{17}}+{\tfrac {1}{19}}\right)+\dots }
ਫਰਮਾ:OEIS2C [0; 1, 6, 1, 2, 1, 2, 956, 3, 1, 1] 0.870588379975 0.63661 97723 67581 34307
Buffon constant 2 π {\displaystyle {\frac {2}{\pi }}} 2 2 ⋅ 2 + 2 2 ⋅ 2 + 2 + 2 2 ⋯ {\displaystyle {\frac {\sqrt {2}}{2}}\cdot {\frac {\sqrt {2+{\sqrt {2}}}}{2}}\cdot {\frac {\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}{2}}\cdots } Viète product T ਫਰਮਾ:OEIS2C [0;1,1,1,3,31,1,145,1,4,2,8,1,6,1,2,3,1,4,...] 1540 to 1603 0.63661977236758134307553505349005745 0.59634 73623 23194 07434 Euler–Gompertz constant G {\displaystyle {G}} ∫ 0 ∞ e − n 1 + n d n = ∫ 0 1 1 1 − ln n d n = 1 1 + 1 1 + 1 1 + 2 1 + 2 1 + 3 1 + 3 / ⋯ {\displaystyle \!\int \limits _{0}^{\infty }\!\!{\frac {e^{-n}}{1{+}n}}\,dn=\!\!\int \limits _{0}^{1}\!\!{\frac {1}{1{-}\ln n}}\,dn=\textstyle {\tfrac {1}{1+{\tfrac {1}{1+{\tfrac {1}{1+{\tfrac {2}{1+{\tfrac {2}{1+{\tfrac {3}{1+3{/\cdots }}}}}}}}}}}}}} integral [ 0 to ∞ ] {( e ^- n ) / ( 1 + n )} I ਫਰਮਾ:OEIS2C [0;1,1,2,10,1,1,4,2,2,13,2,4,1,32,4,8,1,1,1,...] 0.59634736232319407434107849936927937 i ··· Imaginary number i {\displaystyle {i}} − 1 = ln ( − 1 ) π e i π = − 1 {\displaystyle {\sqrt {-1}}={\frac {\ln(-1)}{\pi }}\qquad \qquad \mathrm {e} ^{i\,\pi }=-1} C I 1501 to 1576 i 2.74723 82749 32304 33305 Ramanujan nested radical R 5 {\displaystyle R_{5}} 5 + 5 + 5 − 5 + 5 + 5 + 5 − ⋯ = 2 + 5 + 15 − 6 5 2 {\displaystyle \scriptstyle {\sqrt {5+{\sqrt {5+{\sqrt {5-{\sqrt {5+{\sqrt {5+{\sqrt {5+{\sqrt {5-\cdots }}}}}}}}}}}}}}\;=\textstyle {\frac {2+{\sqrt {5}}+{\sqrt {15-6{\sqrt {5}}}}}{2}}} ( 2 + sqrt ( 5 ) + sqrt ( 15 -6 sqrt ( 5 ))) / 2 A [2;1,2,1,21,1,7,2,1,1,2,1,2,1,17,4,4,1,1,4,2,...] 2.74723827493230433305746518613420282 0.56714 32904 09783 87299 Omega constant, Lambert W function Ω {\displaystyle {\Omega }} ∑ n = 1 ∞ ( − n ) n − 1 n ! = ( 1 e ) ( 1 e ) ⋅ ⋅ ( 1 e ) = e − Ω = e − e − e ⋅ ⋅ − e {\displaystyle \sum _{n=1}^{\infty }{\frac {(-n)^{n-1}}{n!}}=\,\left({\frac {1}{e}}\right)^{\left({\frac {1}{e}}\right)^{\cdot ^{\cdot ^{\left({\frac {1}{e}}\right)}}}}=e^{-\Omega }=e^{-e^{-e^{\cdot ^{\cdot ^{-e}}}}}} Sum [ n = 1 to ∞ ] {( - n ) ^ ( n -1 ) / n ! } T ਫਰਮਾ:OEIS2C [0;1,1,3,4,2,10,4,1,1,1,1,2,7,306,1,5,1,2,1,...] 0.56714329040978387299996866221035555 0.96894 61462 59369 38048 Beta(3) β ( 3 ) {\displaystyle {\beta }(3)} π 3 32 = ∑ n = 1 ∞ − 1 n + 1 ( − 1 + 2 n ) 3 = 1 1 3 − 1 3 3 + 1 5 3 − 1 7 3 + ⋯ {\displaystyle {\frac {\pi ^{3}}{32}}=\sum _{n=1}^{\infty }{\frac {-1^{n+1}}{(-1+2n)^{3}}}={\frac {1}{1^{3}}}{-}{\frac {1}{3^{3}}}{+}{\frac {1}{5^{3}}}{-}{\frac {1}{7^{3}}}{+}\cdots } Sum [ n = 1 to ∞ ] {( -1 ) ^ ( n + 1 ) / ( -1 + 2 n ) ^ 3 } T ਫਰਮਾ:OEIS2C [0;1,31,4,1,18,21,1,1,2,1,2,1,3,6,3,28,1,...] 0.96894614625936938048363484584691860 2.23606 79774 99789 69640 Square root of 5, Gauss sum 5 {\displaystyle {\sqrt {5}}} ( n = 5 ) ∑ k = 0 n − 1 e 2 k 2 π i n = 1 + e 2 π i 5 + e 8 π i 5 + e 18 π i 5 + e 32 π i 5 {\displaystyle \scriptstyle (n=5)\displaystyle \sum _{k=0}^{n-1}e^{\frac {2k^{2}\pi i}{n}}=1+e^{\frac {2\pi i}{5}}+e^{\frac {8\pi i}{5}}+e^{\frac {18\pi i}{5}}+e^{\frac {32\pi i}{5}}} Sum [ k = 0 to 4 ] { e ^ ( 2 k ^ 2 pi i / 5 )} A ਫਰਮਾ:OEIS2C [2;4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,...] = [2;4 ,...] 2.23606797749978969640917366873127624 3.35988 56662 43177 55317 Prévost constant Reciprocal Fibonacci constant Ψ {\displaystyle \Psi } ∑ n = 1 ∞ 1 F n = 1 1 + 1 1 + 1 2 + 1 3 + 1 5 + 1 8 + 1 13 + ⋯ {\displaystyle \sum _{n=1}^{\infty }{\frac {1}{F_{n}}}={\frac {1}{1}}+{\frac {1}{1}}+{\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{5}}+{\frac {1}{8}}+{\frac {1}{13}}+\cdots } Fn : Fibonacci series
Sum [ n = 1 to ∞ ] { 1 / Fibonacci [ n ]} I ਫਰਮਾ:OEIS2C [3;2,1,3,1,1,13,2,3,3,2,1,1,6,3,2,4,362,...] ? 3.35988566624317755317201130291892717 2.68545 20010 65306 44530 Khinchin's constant K 0 {\displaystyle K_{\,0}} ∏ n = 1 ∞ [ 1 + 1 n ( n + 2 ) ] ln n / ln 2 {\displaystyle \prod _{n=1}^{\infty }\left[{1+{1 \over n(n+2)}}\right]^{\ln n/\ln 2}} Prod [ n = 1 to ∞ ] {( 1 + 1 / ( n ( n + 2 ))) ^ ( ln ( n ) / ln ( 2 ))} T ਫਰਮਾ:OEIS2C [2;1,2,5,1,1,2,1,1,3,10,2,1,3,2,24,1,3,2,...] 1934 2.68545200106530644530971483548179569