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List of mathematical constants

03 Oct 2018 National Radio TV of Afghanistan

A mathematical constant is a number, which has a special meaning for calculations. For example, the constant π means the ratio of the length of a circle’s circumference to its diameter. This value is always the same for any circle.

Tables structure[edit]

Value numerical of the constant and link to MathWorld or to OEIS Wiki.

LaTeX: Formula or series in TeX format.

Formula: For use in program Wolfram Alpha.

OEIS: On-Line Encyclopedia of Integer Sequences.

Continued fraction: In the simple form [to integer; frac1, frac2, frac3, …], overline if periodic.

Year: Discovery of the constant, or dates of the author.

Web format: Value in appropriate format for web browsers.

Nº: Number types.

R – Rational number

I – Irrational number

A – Algebraic number

T – Transcendental number

C – Non-real complex number

Table of constants and functions[edit]

You can choose the order of the list by clicking on the name, value, OEIS, etc..

Value

Name

Graphics

Symbol

LaTeX

Formula

OEIS

Continued fraction

Year

Web format

0.74048 04896 93061 04116 [Mw 1]

Hermite constant Sphere packing 3DKepler conjecture [1]

{\displaystyle {\mu _{_{K}}}}

{\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.[2]

pi/(3 sqrt(2))

 A093825

[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 [3]

{\displaystyle \pi ^{e}}

{\displaystyle \pi ^{e}}

pi^e

 A059850

[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 [Mw 2]

Fransén–Robinson constant [4]

{\displaystyle {F}}

{\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)}]

 A058655

[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 [Mw 3]

Fractal dimension of the Apollonian packing of circles

[5]  · [6]

{\displaystyle \varepsilon }

 A052483

[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 [Mw 4]

Infinite Tetration of i[7]

{\displaystyle {}^{\infty }{i}}

{\displaystyle \lim _{n\to \infty }{}^{n}i=\lim _{n\to \infty }\underbrace {i^{i^{\cdot ^{\cdot ^{i}}}}} _{n}}

i^i^i^i^i^i^...

C

 A077589

 A077590

[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 [Mw 5]

Sum of the reciprocals of the averages of the twin prime pairs, JJGJJG

{\displaystyle B_{1}}

{\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 + ...

 A241560

[0; 1, 13, 19, 4, 2, 3, 1, 1]

2014

0.928835827131

0.63092 97535 71457 43709 [Mw 6]

Fractal dimension of the Cantor set [8]

{\displaystyle d_{f}(k)}

{\displaystyle \lim _{\varepsilon \to 0}{\frac {\log N(\varepsilon )}{\log(1/\varepsilon )}}={\frac {\log 2}{\log 3}}}

log(2)/log(3)

N[3^x=2]

T

 A102525

[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 [Mw 7]

Inverse of Pi, Ramanujan[9]

{\displaystyle {\frac {1}{\pi }}}

{\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 ∞]

{((4n)!/n!^4)

*(1103+ 26390n)

/ 396^(4n)}

T

 A049541

[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 [Mw 8]

Flajolet and Richmond [10]

{\displaystyle {Q}}

{\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)…}

prod[n=1 to ∞]

{1-1/2^n}

 A048651

[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 [Mw 9]

Lieb’s square ice constant [11]

{\displaystyle {W}_{2D}}

{\displaystyle \lim _{n\to \infty }(f(n))^{n^{-2}}=\left({\frac {4}{3}}\right)^{\frac {3}{2}}={\frac {8}{3{\sqrt {3}}}}}

(4/3)^(3/2)

A

 A118273

[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 [Mw 10]

{\displaystyle i^{i}} [12]

{\displaystyle i^{i}}

{\displaystyle e^{-{\frac {\pi }{2}}}}

e^(-π/2)

T

 A049006

[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

{\displaystyle {\alpha }}

{\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 }}}

π/ln(2)

 A163973

[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 [Mw 11]

Hyperbolic tangent of 1 [13]

{\displaystyle {th}\,1}

{\displaystyle -i\tan(i)={\frac {e-{\frac {1}{e}}}{e+{\frac {1}{e}}}}={\frac {e^{2}-1}{e^{2}+1}}}

(e-1/e)/(e+1/e)

T

 A073744

[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 [Mw 12]

Chebyshev constant[14] · [15]

{\displaystyle {\lambda _{Ch}}}

{\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))

 A249205

[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 [Ow 1]

MKB constant

[16] · [17] · [18]

{\displaystyle M_{I}}

{\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_(2n->∞) int[1 to 2n]

{exp(i*Pi*x)*x^(1/x) dx}

C

 A255727

 A255728

[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 [Mw 13]

Cube root of 2

Delian Constant

{\displaystyle {\sqrt[{3}]{2}}}

{\displaystyle {\sqrt[{3}]{2}}}

2^(1/3)

A

 A002580

[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 [Mw 14]

Smarandache Constant 1ª [19]

{\displaystyle {S_{1}}}

{\displaystyle \sum _{n=2}^{\infty }{\frac {1}{\mu (n)!}}{\color {white}….\color {black}}} where μ(n) is the Kempner function

 A048799

[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

{\displaystyle {{F}_{CG}}_{(i)}}

{\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

 A156590

 A156548

[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 [Mw 15][Ow 2]

Double factorial

constant

{\displaystyle {C_{_{n!!}}}}

{\displaystyle \sum _{n=0}^{\infty }{\frac {1}{n!!}}={\sqrt {e}}\left[{\frac {1}{\sqrt {2}}}+\gamma ({\tfrac {1}{2}},{\tfrac {1}{2}})\right]}

Sum[n=0 to ∞]{1/n!!}

 A143280

[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 [Mw 16]

Madelung Constant 2[20]

{\displaystyle {H}_{2}(2)}

{\displaystyle \pi \ln(3){\sqrt {3}}}

Pi Log[3]Sqrt[3]

 A086055

[5;1,44,2,2,1,15,1,1,12,1,65,11,1,3,1,1,…]

5.97798681217834912266905331933922774

0.91893 85332 04672 74178 [Mw 17]

Raabe’s formula [21]

{\displaystyle {\zeta ‘(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

 A075700

[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 [Mw 18]

Lower limit in the moving sofa problem[22]

{\displaystyle {S_{_{H}}}}

{\displaystyle {\frac {\pi }{2}}+{\frac {2}{\pi }}}

pi/2 + 2/pi

T

 A086118

[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 [Mw 19]

Salem number,[23]Lehmer’s conjecture

{\displaystyle {\sigma _{_{10}}}}

{\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

 A073011

[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 [Mw 20]

Artin constant [24]

{\displaystyle {C}_{Artin}}

{\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))}

 A005596

[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 [Mw 21]

Volume of Reuleaux tetrahedron [25]

{\displaystyle {V_{_{R}}}}

{\displaystyle {\frac {s^{3}}{12}}(3{\sqrt {2}}-49\,\pi +162\,\arctan {\sqrt {2}})}

(3*Sqrt[2] - 49*Pi + 162*ArcTan[Sqrt[2]])/12

 A102888

[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 [Mw 22]

Murata Constant [26]

{\displaystyle {C_{m}}}

{\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}

 A065485

[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 [Mw 23]

Paris Constant

{\displaystyle C_{Pa}}

{\displaystyle \prod _{n=2}^{\infty }{\frac {2\varphi }{\varphi +\varphi _{n}}}\;,\;\varphi {=}{\frac {1+{\sqrt {5}}}{2}}}  with  {\displaystyle \varphi _{n}{=}{\sqrt {1{+}\varphi _{n{-}1}}}}   and   {\displaystyle \varphi _{1}{=}1}

 A105415

[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 [Mw 24]

Radians

Golden angle [27]

{\displaystyle {b}}

{\displaystyle (4-2\,\Phi )\,\pi =(3-{\sqrt {5}})\,\pi } = 137.5077640500378546 …°

(4-2*Phi)*Pi

T

 A131988

[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 [Mw 25]

Lebesgue constant L2 [28]

{\displaystyle {L2}}

{\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

 A226655

[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 [Mw 26]

Vardi constant[29]

{\displaystyle {V_{c}}}

{\displaystyle {\frac {\sqrt {3}}{\sqrt {2}}}\prod _{n\geq 1}\left(1+{1 \over (2e_{n}-1)^{2}}\right)^{\!1/2^{n+1}}}

 A076393

[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[Mw 27]

Area of the Mandelbrot fractal [30]

{\displaystyle \gamma }

This is conjectured to be: {\displaystyle {\sqrt {6\pi -1}}-e=1.506591651\cdots }

 A098403

[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 [Mw 28]

183213 ones

Exponential factorialconstant

{\displaystyle {S_{Ef}}}

{\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

 A080219

[1; 1, 1, 1, 1, 2, 1, 808, 2, 1, 2, 1, 14,…]

1.61111492580837673611111111111111111

1.11786 41511 89944 97314 [Mw 29]

Goh-Schmutz constant [31]

{\displaystyle C_{GS}}

{\displaystyle \int _{0}^{\infty }{\frac {\log(s+1)}{e^{s}-1}}\ ds=\!-\!\sum _{n=1}^{\infty }{\frac {e^{n}}{n}}Ei(-n)}

Ei: Exponential Integral

Integrate{

log(s+1)

/(E^s-1)}

 A143300

[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 [Ow 3]

Fixed points

Super-Logarithm[32] ·Tetration

{\displaystyle {-W(-1)}}

{\displaystyle \lim _{n\rightarrow \infty }}{\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 {\textstyle 0,1,e,e^{e},e^{e^{e}}}, etc.

-W(-1)

where W=ProductLog

Lambert W function

C

 A059526

 A059527

[-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 [Mw 30]

Bernstein’s constant[33]

{\displaystyle {\beta }}

{\displaystyle \approx {\frac {1}{2{\sqrt {\pi }}}}}

1/(2 sqrt(pi))

T

 A073001

[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 [Mw 31]

Twin Primes Constant[34]

{\displaystyle {C}_{2}}

{\displaystyle \prod _{p=3}^{\infty }{\frac {p(p-2)}{(p-1)^{2}}}}

prod[p=3 to ∞]

{p(p-2)/(p-1)^2

 A005597

[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 [Mw 32]

Fibonacci Factorial constant [35]

{\displaystyle F}

{\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}

 A062073

[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 [Mw 33]

Kepler–Bouwkamp constant [36]

{\displaystyle {\rho }}

{\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)}

 A085365

[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 [Mw 34]

Komornik–Loreti constant [37]

{\displaystyle {q}}

{\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

 A055060

[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 [Mw 35]

Bronze ratio [38]

{\displaystyle {\sigma }_{\,Rr}}

{\displaystyle {\frac {3+{\sqrt {13}}}{2}}=1+{\sqrt {3+{\sqrt {3+{\sqrt {3+{\sqrt {3+\cdots }}}}}}}}}

(3+sqrt 13)/2

A

 A098316

[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 [Mw 36]

Disk Covering [39]

{\displaystyle {C_{5}}}

{\displaystyle {\frac {1}{\sum \limits _{n=0}^{\infty }{\frac {1}{\binom {3n+2}{2}}}}}={\frac {3{\sqrt {3}}}{2\pi }}}

3 Sqrt[3]/(2 Pi)

T

 A086089

[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 [Mw 37]

Gelfond–Schneider constant [40]

{\displaystyle G_{\,GS}}

{\displaystyle 2^{\sqrt {2}}}

2^sqrt{2}

T

 A007507

[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 [Mw 38]

Khinchin-Lévy constant [41]

{\displaystyle \gamma }

{\displaystyle e^{\pi ^{2}/(12\ln 2)}}

e^(\pi^2/(12 ln(2))

 A086702

[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 [Mw 39]

Chi Function

Hyperbolic cosine integral

{\displaystyle {\operatorname {Chi()} }}

{\displaystyle \gamma +\int _{0}^{x}{\frac {\cosh t-1}{t}}\,dt}{\displaystyle \scriptstyle \gamma \,{\text{= Euler–Mascheroni constant= 0.5772156649…}}}

Chi(x)

 A133746

[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 [Mw 40]

Viswanath constant[42]

{\displaystyle {C}_{Vi}}

{\displaystyle \lim _{n\to \infty }|a_{n}|^{\frac {1}{n}}}      where an = Fibonacci sequence

lim_(n->∞)

|a_n|^(1/n)

T ?

 A078416

[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 [Mw 41]

Favard constant [43]

{\displaystyle {\tfrac {3}{4}}\zeta (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/((2n-1)^2)}

T

 A111003

[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

{\displaystyle {\sqrt {2\pi }}}

{\displaystyle {\sqrt {2\pi }}=\lim _{n\to \infty }{\frac {n!\;e^{n}}{n^{n}{\sqrt {n}}}}{\color {white}….\color {black}}} Stirling’s approximation

sqrt (2 pi)

T

 A019727

[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

{\displaystyle {\sqrt {\tau e}}}

{\displaystyle {\sqrt {2\pi e}}}

sqrt(2 pi e)

 A019633

[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 [Mw 42]

Lochs constant [44]

{\displaystyle {{\text{£}}_{_{Lo}}}}

{\displaystyle {\frac {6\ln 2\ln 10}{\pi ^{2}}}}

6*ln(2)*ln(10)/Pi^2

 A086819

[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 [Mw 43]

Area bounded by the

eccentric rotation of

Reuleaux triangle [45]

{\displaystyle {\mathcal {T}}_{R}}

{\displaystyle a^{2}\cdot \left(2{\sqrt {3}}+{\frac {\pi }{6}}-3\right)}    where a= side length of the square

2 sqrt(3)+pi/6-3

T

 A066666

[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[46]

{\displaystyle {\mathcal {C}}_{2}}

{\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))}]

 A065463

[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 [Mw 44]

Connective constant[47][48]

{\displaystyle {\mu }}

{\displaystyle {\sqrt {2+{\sqrt {2}}}}\;=\lim _{n\rightarrow \infty }c_{n}^{1/n}}as a root of the polynomial {\displaystyle :\;x^{4}-4x^{2}+2=0}

sqrt(2+sqrt(2))

A

 A179260

[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 [Mw 45]

Gauss–Kuzmin–Wirsing constant [49]

{\displaystyle {\lambda }_{2}}

{\displaystyle \lim _{n\to \infty }{\frac {F_{n}(x)-\ln(1-x)}{(-\lambda )^{n}}}=\Psi (x),}where {\displaystyle \Psi (x)} is an analytic function with {\displaystyle \Psi (0)\!=\!\Psi (1)\!=\!0}.

 A038517

[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 [Mw 46]

Favard constant K1

Wallis product [50]

{\displaystyle {\frac {\pi }{2}}}

{\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 ∞]

{(4n^2)/(4n^2-1)}

T

 A069196

[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 [Mw 47]

Erdős–Borwein constant[51][52]

{\displaystyle {E}_{\,B}}

{\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

 A065442

[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 [Mw 48]

Phi, Golden ratio [53]

{\displaystyle {\varphi }}

{\displaystyle {\frac {1+{\sqrt {5}}}{2}}={\sqrt {1+{\sqrt {1+{\sqrt {1+{\sqrt {1+\cdots }}}}}}}}}

(1+5^(1/2))/2

A

 A001622

[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 [Mw 49]

Riemann Function Zeta(2)

{\displaystyle {\zeta }(\,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 }

Sum[n=1 to ∞]

{1/n^2}

T

 A013661

[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 [Mw 50]

Theodorus constant[54]

{\displaystyle {\sqrt {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

 A002194

[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 [Mw 51]

Kasner number

{\displaystyle {R}}

{\displaystyle {\sqrt {1+{\sqrt {2+{\sqrt {3+{\sqrt {4+\cdots }}}}}}}}}

Fold[Sqrt[#1+#2]

&,0,Reverse

[Range[20]]]

 A072449

[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 [Mw 52]

Universal parabolic constant [55]

{\displaystyle {P}_{\,2}}

{\displaystyle \ln(1+{\sqrt {2}})+{\sqrt {2}}\;=\;\operatorname {arcsinh} (1)+{\sqrt {2}}}

ln(1+sqrt 2)+sqrt 2

T

 A103710

[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 [Mw 53]

Silverman constant[56]

{\displaystyle {{\mathcal {S}}_{_{m}}}}

{\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, σ1() = Divisor function.

Sum[n=1 to ∞]

{1/[EulerPhi(n)

DivisorSigma(1,n)]}

 A093827

[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 [Mw 54]

Area of the regular hexagon with side equal to 1 [57]

{\displaystyle {\mathcal {A}}_{6}}

{\displaystyle {\frac {3{\sqrt {3}}}{2}}}

3 sqrt(3)/2

A

 A104956

[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 [Mw 55]

Feller–Tornier constant [58]

{\displaystyle {{\mathcal {C}}_{_{FT}}}}

{\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 ?

 A065493

[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 [Mw 56]

Baxter’s

Four-coloring

constant [59]

Mapamundi Four-Coloring

{\displaystyle {\mathcal {C}}^{2}}

{\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

3×Gamma(1/3)

^3/(4 pi^2)

 A224273

[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 [Mw 57]

Tetranacci constant

{\displaystyle {\mathcal {T}}}

Positive root of {\displaystyle :\;\;x^{4}-x^{3}-x^{2}-x-1=0}

Root[x+x^-4-2=0]

A

 A086088

[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 [Mw 58]

DeVicci’s tesseract constant

{\displaystyle {f_{(3,4)}}}

The largest cube that can pass through in an 4D hypercube.Positive root of {\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

 A243309

[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 [Mw 59]

Niven’s constant [60]

{\displaystyle {C}}

{\displaystyle 1+\sum _{n=2}^{\infty }\left(1-{\frac {1}{\zeta (n)}}\right)}

1+ Sum[n=2 to ∞]

{1-(1/Zeta(n))}

 A033150

[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 [Mw 60]

Relationship among the area of an equilateral triangle and the inscribed circle.

{\displaystyle {\frac {\pi }{3{\sqrt {3}}}}}

{\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

 A073010

[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 [Mw 61]

Hermite constant [61]

{\displaystyle \gamma _{_{2}}}

{\displaystyle {\frac {2}{\sqrt {3}}}={\frac {1}{\cos \,({\frac {\pi }{6}})}}}

2/sqrt(3)

A

1+

 A246724

[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 [Mw 62]

Prouhet–Thue–Morse constant [62]

{\displaystyle \tau }

{\displaystyle \sum _{n=0}^{\infty }{\frac {t_{n}}{2^{n+1}}}}    where {\displaystyle {t_{n}}} is the Thue–Morse sequence  and

Where {\displaystyle \tau (x)=\sum _{n=0}^{\infty }(-1)^{t_{n}}\,x^{n}=\prod _{n=0}^{\infty }(1-x^{2^{n}})}

T

 A014571

[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 [Mw 63]

Pell constant [63]

{\displaystyle {{\mathcal {P}}_{_{Pell}}}}

{\displaystyle 1-\prod _{n=0}^{\infty }\left(1-{\frac {1}{2^{2n+1}}}\right)}

N[1-prod[n=0 to ∞]

{1-1/(2^(2n+1)}]

T ?

 A141848

[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 [Mw 64]

Laplace limit [64]

{\displaystyle {\lambda }}

{\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

 A033259

[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 [Mw 65]

Hall-Montgomery Constant [65]

{\displaystyle {{\delta }_{_{0}}}}

{\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]]

 A143301

[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 [Mw 66]

Calabi triangleconstant [66]

{\displaystyle {C_{_{CR}}}}

{\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[

2x^3-2x^2-3x+2

==0, {x, 1.5},

WorkingPrecision->40]

A

 A046095

[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 [Mw 67]

Gamma(3/4) [67]

{\displaystyle \Gamma ({\tfrac {3}{4}})}

{\displaystyle \left(-1+{\frac {3}{4}}\right)!=\left(-{\frac {1}{4}}\right)!}

(-1+3/4)!

 A068465

[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 [Mw 68]

Apéry’s constant [68]

{\displaystyle \zeta (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 =}{\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}}}

Sum[n=1 to ∞]

{1/n^3}

I

 A010774

[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 [Mw 69]

Catalan’s constant[69][70][71]

{\displaystyle {C}}

{\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/(2n+1)^2}

T

 A006752

[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 [Mw 70]

Beta(1) [72]

{\displaystyle {\beta }(1)}

{\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/(2n+1)}

T

 A003881

[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 [Mw 71]

Heath-Brown–Moroz constant[73]

{\displaystyle {C_{_{HBM}}}}

{\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 ?

 A118228

[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

Infinite

Tetration of i

{\displaystyle |{}^{\infty }{i}|}

{\displaystyle \lim _{n\to \infty }\left|{}^{n}i\right|=\left|\lim _{n\to \infty }\underbrace {i^{i^{\cdot ^{\cdot ^{i}}}}} _{n}\right|}

Mod(i^i^i^...)

 A212479

[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 [Mw 72]

Sophomore’s dream1

J.Bernoulli [74]

{\displaystyle {I}_{1}}

{\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}

 A083648

[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 [Mw 73]

Sophomore’s dream2

J.Bernoulli [75]

{\displaystyle {I}_{2}}

{\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 }

Sum[n=1 to ∞]

{1/(n^n)}

 A073009

[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 [Mw 74]

Primorial constant

Sum of the product of inverse of primes [76]

{\displaystyle {P_{\#}}}

{\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

 A064648

[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 [Mw 75]

Plouffe’s gamma constant [77]

{\displaystyle {C}}

{\displaystyle {\frac {1}{\pi }}\arctan {\frac {1}{2}}={\frac {1}{\pi }}\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2^{2n+1})(2n+1)}}}

{\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)}

Arctan(1/2)/pi

T

 A086203

[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 [Mw 76]

Plouffe’s A constant [78]

{\displaystyle {A}}

{\displaystyle {\frac {1}{2\pi }}}

1/(2 pi)

T

 A086201

[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 [Mw 77]

Dimer constant 2D,

Domino tiling[79][80]

{\displaystyle {\frac {C}{\pi }}}C=Catalan

{\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}]

 A143233

[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)[81]

{\displaystyle {i}\,!}

{\displaystyle \Gamma (1+i)=i\,\Gamma (i)=\int \limits _{0}^{\infty }{\frac {t^{i}}{e^{t}}}\mathrm {d} t}

Integral_0^∞

t^i/e^t dt

C

 A212877

 A212878

[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 [Mw 78]

Wallis Constant

{\displaystyle W}

{\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

 A007493

[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 [Mw 79]

Sarnak constant

{\displaystyle {C_{sa}}}

{\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 ?

 A065476

[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 [Mw 80]

Time constant [82]

{\displaystyle {\tau }}

{\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}}=}

{\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

 A068996

[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 [83]

{\displaystyle F_{1}}

{\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))]

 A213080

[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 [Mw 81]

Lemniscate Constant [84]

{\displaystyle 2\varpi }

{\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 ]

 A064853

[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 [Mw 82]

Robbins constant [85]

{\displaystyle \Delta (3)}

{\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

 A073012

[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 [Mw 83]

Conway constant [86]

{\displaystyle {\lambda }}

{\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

 A014715

[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 [Mw 84]

Khinchin–Lévy constant[87]

{\displaystyle {\beta }}

{\displaystyle {\frac {\pi ^{2}}{12\,\ln 2}}}

pi^2 /(12 ln 2)

 A100199

[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 [88]

{\displaystyle \beta _{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))/(3n+1)}

 A113476

[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 [Mw 85]

Kempner Serie(0) [89]

{\displaystyle {K_{0}}}

{\displaystyle 1{+}{\frac {1}{2}}{+}{\frac {1}{3}}{+}\cdots {+}{\frac {1}{9}}{+}{\frac {1}{11}}{+}\cdots {+}{\frac {1}{19}}{+}{\frac {1}{21}}{+}\cdots }{\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+...

 A082839

[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 [Mw 86]

Lebesgue constant[90]

{\displaystyle {C_{1}}}

{\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)]

 A243277

[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 [Mw 87]

2nd du Bois-Reymond constant[91]

{\displaystyle {C_{2}}}

{\displaystyle {\frac {e^{2}-7}{2}}=\int _{0}^{\infty }\left|{{\frac {d}{dt}}\left({\frac {\sin t}{t}}\right)^{n}}\right|\,dt-1}

(e^2-7)/2

T

 A062546

[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 [Mw 88]

Lüroth constant[92]

{\displaystyle C_{L}}

{\displaystyle \sum _{n=2}^{\infty }{\frac {\ln \left({\frac {n}{n-1}}\right)}{n}}}

Sum[n=2 to ∞]

log(n/(n-1))/n

 A085361

[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 [Mw 89]

Foias constant α [93]

{\displaystyle F_{\alpha }}

{\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 x1 = αthen the sequence diverges to ∞. When x1 = α, {\displaystyle \,\lim _{n\to \infty }x_{n}{\tfrac {\log n}{n}}=1}

 A085848

[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 [Mw 90]

Foias constant β

{\displaystyle F_{\beta }}

{\displaystyle x^{x+1}=(x+1)^{x}}

x^(x+1)

= (x+1)^x

 A085846

[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 [Mw 91]

Nielsen–Ramanujanconstant [94]

{\displaystyle {\frac {{\zeta }(2)}{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

 A072691

[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 [Mw 92]

Natural logarithm of 2[95]

{\displaystyle Ln(2)}

{\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

 A002162

[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 [Mw 93]

Weierstrass constant [96]

{\displaystyle \sigma ({\tfrac {1}{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)

 A094692

[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 [Mw 94]

Euler–Mascheroni constant

{\displaystyle {\gamma }}

{\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)}

{\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)}

 A001620

[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[97]

{\displaystyle \beta }

{\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))

 A242710

[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 [Mw 95]

Golden spiral

{\displaystyle c}

{\displaystyle \varphi ^{\frac {2}{\pi }}=\left({\frac {1+{\sqrt {5}}}{2}}\right)^{\frac {2}{\pi }}}

GoldenRatio^(2/pi)

 A212224

[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 [Mw 96]

Stephens constant[98]

{\displaystyle C_{S}}

{\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 ?

 A065478

[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 [Mw 97]

Dottie number [99]

{\displaystyle d}

{\displaystyle \lim _{x\to \infty }\cos ^{[x]}(c)=\lim _{x\to \infty }\underbrace {\cos(\cos(\cos(\cdots (\cos(c)))))} _{x}}

cos(c)=c

T

 A003957

[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 [Mw 98]

Taniguchi constant[100]

{\displaystyle C_{T}}

{\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)}{\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 ?

 A175639

[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 [Mw 99]

Gauss’ Lemniscate constant[101]

{\displaystyle L{\text{/}}{\sqrt {2}}}

{\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 }}}}{\displaystyle \scriptstyle \Gamma (){\text{= Gamma function}}}

pi^(3/2)/(2 Gamma(3/4)^2)

 A093341

[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 [102]

{\displaystyle Pr_{1}}

{\displaystyle \prod _{n=2}^{\infty }{\Big (}1+{\frac {1}{n}}{\Big )}^{\frac {1}{n}}}

Prod[n=2 to inf]

{(1+1/n)^(1/n)}

 A242623

[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[103]

{\displaystyle \partial }

{\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))}

 A226317

[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

{\displaystyle S_{5}}

{\displaystyle \displaystyle {\frac {{\sqrt {21}}+1}{2}}=\scriptstyle \,{\sqrt {5+{\sqrt {5+{\sqrt {5+{\sqrt {5+{\sqrt {5+\cdots }}}}}}}}}}}{\displaystyle =1+\,\scriptstyle {\sqrt {5-{\sqrt {5-{\sqrt {5-{\sqrt {5-{\sqrt {5-\cdots }}}}}}}}}}}

(sqrt(21)+1)/2

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 [Mw 100]

Square root of i [104]

{\displaystyle {\sqrt {i}}}

{\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)}

(1+i)/(sqrt 2)

C A

 A010503

[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 [Mw 101]

Alladi–Grinstead constant [105]

{\displaystyle {{\mathcal {A}}_{AG}}}

{\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}

 A085291

[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 [Mw 102]

Sierpiński’s constant[106]

{\displaystyle {K}}

{\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 )}{\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]]

 A062089

[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 [Ow 4]

Reciprocal of the Euler–Mascheroni constant

{\displaystyle {\frac {1}{\gamma }}}

{\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))

 A098907

[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 [Mw 103]

Lebesgue constant (interpolation)[107][108]

{\displaystyle {L_{1}}}

{\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 }}}

1/3 + 2*sqrt(3)/pi

T

 A226654

[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 [Mw 104]

Silver root

Tutte–Beraha constant [109]

{\displaystyle \varsigma }

{\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 }}}}}}}}}

2+2 cos(2Pi/7)

A

 A116425

[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 [Mw 105]

Euler Totient

constant [110][111]

{\displaystyle ET}

{\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}}}}

zeta(2)*zeta(3)

/zeta(6)

 A082695

[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 [112]

{\displaystyle {\sqrt[{4}]{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

 A011003

[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 [Mw 106]

Area of Ford circle[113]

{\displaystyle A_{CF}}

{\displaystyle \sum _{q\geq 1}\sum _{(p,q)=1 \atop 1\leq p2}{{\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]

 A084945

[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 [Mw 154]

Gelfond’s constant[179]

{\displaystyle {e}^{\pi }}

{\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

 A039661

[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 [180]

{\displaystyle e^{2}}

{\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 }

Sum[n=0 to ∞]

{2^n/n!}

T

 A072334

[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 [Mw 155]

Hafner–Sarnak–McCurley constant(1) [181]

{\displaystyle {\sigma }}

{\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}

 A085849

[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 [Mw 156]

Hafner–Sarnak–McCurley constant(2) [182]

{\displaystyle {\frac {1}{\zeta (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

 A059956

[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 [Mw 157]

Champernowne constant [183]

{\displaystyle C_{10}}

{\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

 A033307

[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 [Mw 158]

Landau–Ramanujan constant [184]

{\displaystyle K}

{\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 ?

 A064533

[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 [Mw 159]

Number e, Euler’s number [185]

{\displaystyle {e}}

{\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

 A001113

[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 [Mw 160]

Inverse of Number e[186]

{\displaystyle {\frac {1}{e}}}

{\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

 A068985

[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 [187]

{\displaystyle {H}_{2n+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 …

 A242760

[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 [188]

{\displaystyle {H}_{2n}}

{\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 [Mw 161]

π number, Archimedes number[189]

{\displaystyle \pi }

{\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/(2n+1)}

T

 A000796

[3;7,15,1,292,1,1,1,2,1,3,1,14,2,1,1,2,2,2,…]

3.14159265358979323846264338327950288

1.92878 00… [Mw 162]

Wright constant [190]

{\displaystyle {\omega }}

{\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 }

 A086238

[1; 1, 13, 24, 2, 1, 1, 3, 1, 1, 3]

1.9287800…

0.46364 76090 00806 11621

Machin–Gregory series[191]

{\displaystyle \arctan {\frac {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)^(2n+1)

/(2n+1)}

I

 A073000

[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 [Mw 163]

Continued fraction constant, Bessel function[192]

{\displaystyle {C}_{CF}}

{\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

 A052119

[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 [Mw 164]

Brun 2 constant = Σ inverse of Twin primes[193]

{\displaystyle {B}_{\,2}}

{\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 }

 A065421

[1; 1, 9, 4, 1, 1, 8, 3, 4, 4, 2, 2]

1.902160583104

0.87058 83799 75 [Mw 165]

Brun 4 constant = Σ inv.prime quadruplets[194]

{\displaystyle {B}_{\,4}}

{\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}}}}{\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 }

 A213007

[0; 1, 6, 1, 2, 1, 2, 956, 3, 1, 1]

0.870588379975

0.63661 97723 67581 34307 [Mw 166][Ow 8]

Buffon constant[195]

{\displaystyle {\frac {2}{\pi }}}

{\displaystyle {\frac {\sqrt {2}}{2}}\cdot {\frac {\sqrt {2+{\sqrt {2}}}}{2}}\cdot {\frac {\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}{2}}\cdots } Viète product

2/Pi

T

 A060294

[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 [Mw 167]

Euler–Gompertz constant [196]

{\displaystyle {G}}

{\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

 A073003

[0;1,1,2,10,1,1,4,2,2,13,2,4,1,32,4,8,1,1,1,…]

0.59634736232319407434107849936927937

i ··· [Mw 168]

Imaginary number[197]

{\displaystyle {i}}

{\displaystyle {\sqrt {-1}}={\frac {\ln(-1)}{\pi }}\qquad \qquad \mathrm {e} ^{i\,\pi }=-1}

sqrt(-1)

C I

1501

to

1576

i

2.74723 82749 32304 33305

Ramanujan nested radical [198]

{\displaystyle R_{5}}

{\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 [Mw 169]

Omega constant, Lambert W function[199]

{\displaystyle {\Omega }}

{\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

 A030178

[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) [200]

{\displaystyle {\beta }(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+2n)^3}

T

 A153071

[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 [201]

{\displaystyle {\sqrt {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^(2k^2 pi i/5)}

A

 A002163

[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 [Mw 170]

Prévost constant Reciprocal Fibonacci constant[202]

{\displaystyle \Psi }

{\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

 A079586

[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 [Mw 171]

Khinchin’s constant[203]

{\displaystyle K_{\,0}}

{\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

 A002210

[2;1,2,5,1,1,2,1,1,3,10,2,1,3,2,24,1,3,2,…]

1934

2.68545200106530644530971483548179569

See also[edit]

Invariant (mathematics)

List of numbers

Mathematical constant

Original Link: https://rta.org.af/eng/2018/10/03/list-of-mathematical-constants/

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