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# Mathematical constants and functions

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### Mathematical constants and functions

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

• 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.
• : Number types.

## Table of constants and functions

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
0.20787 95763 50761 90854 [Mw 1] i^i [1]

{i}^{i} e^ \frac{-\pi}{2}
 e^(-π/2)

T [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 {\alpha} \frac{\pi}{ln(2)} = \frac{\sum_{n = 0}^\infty \frac{4(-1)^n}{2n+1}} {\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n}} = \frac{\frac{4}{1} {-} \frac{4}{3} {+} \frac{4}{5} {-} \frac{4}{7} {+} \frac{4}{9} - ...} {\frac{1}{1}{-}\frac{1}{2}{+}\frac{1}{3}{-}\frac{1}{4}{+}\frac{1}{5}-...}
 π/ln(2)

[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 2] Hyperbolic tangent of 1 [2] {th} \, 1 -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 [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 3] Chebyshev constant [3] · [4]

{\lambda_{Ch}} \frac{\Gamma(\tfrac14)^2}{4 \pi^{3/2}} = \frac{4 (\tfrac14 !)^2}{\pi^{3/2}}
(Gamma(1/4)^2)
/(4 pi^(3/2))

[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
[5] · [6] · [7]
M_I \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
[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 4] Cube root of 2
Delian Constant
\sqrt[3]{2} \sqrt[3]{2}
2^(1/3)

A [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 5] Smarandache Constant 1ª [8] {S_1} \sum_{n=2}^\infty \frac1{\mu(n)!} {\color{white}....\color{black}} where μ(n) is the Kempner function [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
_{(i)} \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(\tfrac12 {+} \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

[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 6] [Ow 2] Double factorial
constant
{C_{_{n!!}}} \sum_{n=0}^{\infty} \frac{1}{n!!} = \sqrt{e} \left[\frac {1}{\sqrt{2}}+\gamma(\tfrac12 ,\tfrac12)\right]
Sum[n=0 to ∞]{1/n!!}

[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 7] Madelung Constant 2 [9]

{H}_{2}(2) \pi \ln(3) \sqrt 3
Pi Log[3]Sqrt[3]

[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 8] Raabe's formula [10]

{\zeta'(0)} \int\limits_a^{a+1}\log\Gamma(t)\,\mathrm dt = \tfrac12\log2\pi + a\log a - a,\quad a \ge 0
integral_a^(a+1)
{log(Gamma(x))+a-a log(a)} dx

[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 9] Lower limit in the moving sofa problem [11] {S_{_{H}}} \frac {\pi}{2} +\frac {2}{\pi} \,
pi/2 + 2/pi

T [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 10] Salem number,[12]

{\sigma_{_{10}}} 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 [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 11] Artin constant [13] {C}_{Artin} \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))}

[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 12] Volume of Reuleaux tetrahedron [14] {V_{_{R}}} \frac{s^3}{12}(3\sqrt2 - 49 \, \pi + 162 \, \arctan\sqrt2)
(3*Sqrt[2] - 49*Pi + 162*ArcTan[Sqrt[2]])/12

[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 13] Murata Constant [15] {C_m} \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}

[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 14] Paris Constant C_{Pa} \prod_{n=2}^\infty \frac{2 \varphi}{\varphi+ \varphi_n} \; ,\; \varphi {=} {Fi}  con  \varphi_n {=} \sqrt{1 {+} \varphi_{n {-} 1}}   y   \varphi_1 {=} 1 [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 15]
Golden angle [16] {b} (4-2\,\Phi)\,\pi = (3-\sqrt{5})\,\pi = 137.5077640500378546 ...°
(4-2*Phi)*Pi

T [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 16] Lebesgue constant L2 [17]

{L2} \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})} \,d t
1/5 + sqrt(25 -
2*sqrt(5))/Pi

T [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 17] Vardi constant[18]

{V_c} \frac{\sqrt{3}}{\sqrt{2}}\prod_{n\ge1}\left(1+{1\over(2e_n-1)^2}\right)^{\!1/2^{n+1}} [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 18] Area of the Mandelbrot fractal [19] \gamma This is conjectured to be: \sqrt{6\pi -1} - e = 1,506591651... [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 19]
183213 ones
Exponential factorial constant {S_{Ef}} \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 [1; 1, 1, 1, 1, 2, 1, 808, 2, 1, 2, 1, 14,...] 1.61111492580837673611111111111111111
1.11786 41511 89944 97314 [Mw 20] Goh-Schmutz constant [20] C_{GS} \int^\infty_0\frac{\log(s+1)}{e^s-1} \ ds = \! - \! \sum_{n=1}^\infty \frac {e^n}{n} Ei(-n) \overset {Ei:} \underset{ Integral} {\scriptstyle Exponential}
Integrate{
log(s+1)
/(E^s-1)}

[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[21] · Tetration
{-W(-1)} \lim_{n\rightarrow \infty} 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), etc.

-W(-1)

where W=ProductLog
Lambert W function

C
[-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 21] Bernstein's constant [22]

{\beta} \approx \frac {1}{2\sqrt {\pi}}
1/(2 sqrt(pi))

T [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 22] Twin Primes Constant [23]

{C}_{2} \prod_{p=3}^\infty \frac{p(p-2)}{(p-1)^2}
prod[p=3 to ∞]
{p(p-2)/(p-1)^2

[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 23] Fibonacci Factorial constant [24] F \prod_{n = 1}^\infty \left(1 - \left( -\frac{1} \!\right ) \! {+} \frac{q{-}2}{q{-}1}=0
FindRoot[(prod[n=0 to ∞]
{1-1/(x^2^n)}+(x-2)
/(x-1))= 0, {x, 1.7},
WorkingPrecision->30]

T [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 26] Bronze ratio [27]

{\sigma}_{\,Rr} \frac {3+\sqrt{13}}{2} = 1+\sqrt{3+\sqrt{3+\sqrt{3+\sqrt{3+\cdots}}}}
(3+sqrt 13)/2

A [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 27] Disk Covering [28] {C_5} {\frac{1}}}} = \frac{3\sqrt{3}}{2\pi}
3 Sqrt[3]/(2 Pi)

T [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 28] Gelfond–Schneider constant [29] G_{\,GS} 2^{\sqrt{2}}
2^sqrt{2}

T [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 29] Khinchin-Lévy constant [30] \gamma e^{\pi^2/(12\ln2)}
e^(\pi^2/(12 ln(2))

[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 30] Chi Function
Hyperbolic cosine integral
{\operatorname{Chi()}}
\gamma + \int_0^x\frac{\cosh t-1}{t}\,dt

\scriptstyle \gamma \, \text{= Euler–Mascheroni constant= 0.5772156649...}

Chi(x)

[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 31] Viswanath constant[31] {C}_{Vi} \lim_{n \to \infty}|a_n|^\frac{1}{n}      where an = Fibonacci sequence
lim_(n->∞)
|a_n|^(1/n)

T ? [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 32] Favard constant [32] \tfrac34\zeta(2) \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 [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 \sqrt{2 \pi} \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 [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

\sqrt{\tau e} \sqrt{2 \pi e}
sqrt(2 pi e)

[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 33] Lochs constant [33]

{\text{£}_{_{Lo}}} \frac {6 \ln 2 \ln 10}{ \pi^2}
6*ln(2)*ln(10)/Pi^2

[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 34] Area bounded by the
eccentric rotation of
Reuleaux triangle [34]
\mathcal{T}_R 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 [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 [35]

\mathcal{C}_2 \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))}]

[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 35] Connective constant [36][37] {\mu} \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

sqrt(2+sqrt(2))

A [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 36] Gauss-Kuzmin-Wirsing constant [38] {\lambda}_{2} \lim_{n \to \infty}\frac{F_n(x) - \ln(1 - x)}{(-\lambda)^n} = \Psi(x),

where \Psi(x) is an analytic function with \Psi(0) \!=\! \Psi(1) \!=\! 0.

[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 37] Favard constant K1
Wallis product [39]
{\frac{\pi}{2}} \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 [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 38] Erdős–Borwein constant[40][41]

{E}_{\,B} \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 [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 39] Phi, Golden ratio [42] {\varphi} \frac{1 + \sqrt{5}}{2} = \sqrt{1 + \sqrt{1 + \sqrt{1 + \sqrt{1 + \cdots}}}}
(1+5^(1/2))/2

A [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 40] Riemann Function Zeta(2) {\zeta}(\,2) \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 [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 41] Theodorus constant[43] \sqrt{3} \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 [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 42] Kasner number {R} \sqrt{1 + \sqrt{2 + \sqrt{3 + \sqrt{4 + \cdots}}}}
Fold[Sqrt[#1+#2]
&,0,Reverse
[Range[20]]]

[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 43] Universal parabolic constant [44] {P}_{\,2} \ln(1 + \sqrt2) + \sqrt2 \; = \; \operatorname{arcsinh}(1)+\sqrt{2}
ln(1+sqrt 2)+sqrt 2

T [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 44] Silverman constant[45]

{\mathcal{S}_{_{m}}} \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)]}

[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 45] Area of the regular hexagon with side equal to 1 [46] \mathcal{A}_6 \frac{3 \sqrt{3}}{2}\,l^2
3 sqrt(3)/2

A [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 46] Feller-Tornier
constant [47]

{\mathcal{C}_{_{FT}}} \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 ? [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 47] Baxter's
Four-coloring
constant [48]
Mapamundi Four-Coloring \mathcal{C}^2 \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)

[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 48] Tetranacci constant

\mathcal{T} Positive root of : \;\; x^4-x^3-x^2-x-1=0
Root[x+x^-4-2=0]

A [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 49] DeVicci's tesseract constant {f_{(3,4)}} The largest cube that can pass through in an 4D hypercube.

Positive root of : \;\; 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 [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 50] Niven's constant [49] {C} 1+\sum_{n = 2}^\infty \left(1-\frac{1}{\zeta(n)} \right)
1+ Sum[n=2 to ∞]
{1-(1/Zeta(n))}

[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 51] Relationship among the area of an equilateral triangle and the inscribed circle. \frac{\pi}{3 \sqrt 3}
\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 [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 52] Hermite constant [50] \gamma_{_{2}} \frac{2}{\sqrt{3}} = \frac{1}{\cos \, (\frac{\pi}{6})}
2/sqrt(3)

A 1+
[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 53] Prouhet–Thue–Morse constant [51] \tau \sum_{n=0}^{\infty} \frac{t_n}{2^{n+1}}    where {t_n} is the Thue–Morse sequence  and
Where \tau(x) = \sum_{n=0}^{\infty} (-1)^{t_n} \, x^n = \prod_{n=0}^{\infty} ( 1 - x^{2^n} )
T [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 54] Pell constant [52]

{\mathcal{P}_{_{Pell}}} 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 ? [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 55] Laplace limit [53] {\lambda} \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

[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 56] Hall-Montgomery Constant [54] } 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]]

[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 57] Calabi triangle constant [55] {C_{_{CR}}} {1 \over 3} + {(-23 + 3i \sqrt{237})^{\tfrac13} \over 3 \cdot 2^{\tfrac23}} + {11 \over 3 (2 (-23 + 3i \sqrt{237}))^{\tfrac13}}
FindRoot[
2x^3-2x^2-3x+2
==0, {x, 1.5},
WorkingPrecision->40]

A [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 58] Gamma(3/4) [56]

\Gamma(\tfrac34) \left(-1+\frac{3}{4}\right)! = \left(-\frac{1}{4}\right)!
(-1+3/4)!

[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 59] Apéry's constant [57] \zeta(3) \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=

\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 [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 60] Catalan's constant[58][59][60]

{C} \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 [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 61] Beta(1) [61] {\beta}(1) \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 [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 62] Heath-Brown–Moroz constant[62] {C_{_{HBM}}} \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 ? [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
|{}^\infty {i} | \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^...)

[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 63] Sophomore's dream 1 J.Bernoulli [63] {I}_{1} \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}

[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 64] Sophomore's dream 2 J.Bernoulli [64] {I}_{2} \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)}

[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 65] Primorial constant
Sum of the product of inverse of primes [65]
{P_\#} \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 [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 66] Plouffe's gamma constant [66] \frac{1}{\pi} \arctan {\frac{1}{2}} = \frac{1}{\pi}\sum_{n=0}^\infty \frac {(-1)^n}{(2^{2n+1})(2n+1)}
= \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 [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 67] Plouffe's A constant [67]

{A} \frac{1}{2 \pi}
1/(2 pi)

T [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 68] Dimer constant 2D,
Domino tiling[68][69]
{\frac{C}{\pi}}

C=Catalan

\int\limits_{-\pi}^{\pi} \frac{\cosh^{-1}\left(\frac{\sqrt{\cos(t)+3}}{\sqrt2}\right)}{4\pi}\,dt
N[int[-pi to pi]
{arccosh(sqrt(
cos(t)+3)/sqrt(2))
/(4*Pi)dt}]

[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)[70] {i}\,! \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
[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 69] Wallis Constant W \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 [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 70] Sarnak constant {C_{sa} } \prod_{p>2} \Big(1 - \frac{p+2}{p^3}\Big)
N[prod[k=2 to ∞]
{1-(prime(k)+2)
/(prime(k)^3)}]

T ? [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 71] Time constant [71] {\tau} \lim_{n \to \infty} 1-\frac {!n}{n!}=\lim_{n \to \infty} P(n)= \int_{0}^{1}e^{-x}dx = 1{-}\frac{1}{e} =

\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 [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 [72] F_1 \prod_{n=1}^{\infty} \frac{n!}{\sqrt{2\pi n}\left(\frac{n}{e}\right)^n \sqrt[12]{1+\tfrac1{n}}}
N[prod[n=1 to ∞]
n! /(sqrt(2*Pi*n)
*(n/e)^n *(1+1/n)
^(1/12))]

[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 72] Lemniscate Constant [73]
2\varpi \frac = 4\int^1_0 \frac{dx}{\sqrt{(1-x^2)(2-x^2)}}
Gamma[ 1/4 ]^2
/Sqrt[ 2 Pi ]

[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 73] Robbins constant [74] \Delta(3) \frac{4 \! + \! 17\sqrt2 \! -6 \sqrt3 \! -7\pi}{105} \! + \! \frac{\ln(1 \! + \! \sqrt2)}{5} \! + \! \frac{2\ln(2 \! + \! \sqrt3)}{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

[0;1,1,1,21,1,2,1,4,10,1,2,2,1,3,11,1,331,1,4,...] 1978 0.66170718226717623515583113324841358
1.18656 91104 15625 45282 [Mw 75] Khinchin–Lévy constant[76]

{\beta} \frac {\pi^2}{12\,\ln 2}
pi^2 /(12 ln 2)

[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 [77] \beta_3 \int^1_0 \frac{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)}

[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 76] Kempner Serie(0) [78] {K_0} 1{+}\frac12{+}\frac13{+}\cdots{+}\frac19{+}\frac1{11}{+}\cdots{+}\frac1{19}{+}\frac1{21}{+}\cdots{+}\,\text{etc.}

{+}\frac1{99}{+}\frac1{111}{+}\cdots{+}\frac1{119}{+}\frac1{121}{+}\cdots\;\; \overset {Excluding \; all}{ \underset{ containing \; 0.}{\scriptstyle denominators} }

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+...

[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 77] Lebesgue constant [79] {C_1} \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'(\tfrac12)}{\Gamma(\tfrac12)}\!\!\right)
4/pi^2*[(2
Sum[k=1 to ∞]
{ln(k)/(4*k^2-1)})
-poligamma(1/2)]

[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 78] 2nd du Bois-Reymond constant [80] {C_2} \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 [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 79] Lüroth constant[81]
C_L \sum_{n = 2}^\infty \frac{\ln\left(\frac{n}{n-1}\right)}{n}
Sum[n=2 to ∞]
log(n/(n-1))/n

[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 80] Foias constant α [82]

F_\alpha 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 = α then the sequence diverges to ∞. When x1 = α, \, \lim_{n\to\infty} x_n \tfrac{\log n}{n} = 1

[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 81] Foias constant β F_\beta x^{x+1} = (x+1)^x
x^(x+1)
= (x+1)^x

[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 82] Nielsen-Ramanujan constant [83]

\frac{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 [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 83] Natural logarithm of 2 [84] Ln(2) \sum_{n=1}^\infty \frac{1}{n 2^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 [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 84] Weierstrass constant [85]

\sigma(\tfrac12) \frac{e^{\frac{\pi}{8}}\sqrt{\pi}}{4*2^{3/4} {(\frac {1}{4}!)^2}}
(E^(Pi/8) Sqrt[Pi])
/(4 2^(3/4) (1/4)!^2)

[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 85] Euler–Mascheroni constant {\gamma} \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)

= \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)}

[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[86] \beta 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 i t}} \rfloor dt}}
e^((PolyGamma(1,4/3)
- PolyGamma(1,2/3)
+9)/(4*sqrt(3)*Pi))

[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 86] Golden Spiral c \varphi ^ \frac{2}{\pi} = \left(\frac{1 + \sqrt{5}}{2}\right)^{\frac{2}{\pi}}
GoldenRatio^(2/pi)

[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 87] Stephens constant [87] C_S \prod_{n = 1}^\infty \left(1 - \frac{p}{p^3-1}\right)
Prod[n=1 to ∞]
{1-hprime(n)
/(hprime(n)^3-1)}

T ? [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 88] Dottie number [88] d \lim_{x\to \infty} \cos^(c) = \lim_{x\to \infty} \underbrace{\cos(\cos(\cos(\cdots(\cos(c)))))}_x
cos(c)=c

T [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 89] Taniguchi constant [89] C_T \prod_{n = 1}^\infty \left(1 - \frac{3}{\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}}
\scriptstyle \Gamma() \text{= Gamma function}
pi^(3/2)/(2 Gamma(3/4)^2)

[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 [91] Pr_1 \prod_{n = 2}^\infty \Big(1 + \frac{1}{n}\Big)^\frac{1}{n}
Prod[n=2 to inf]
{(1+1/n)^(1/n)}

[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 [92] \partial \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))}

[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 \frac{\sqrt{21}+1}{2} = \scriptstyle \, \sqrt{5+\sqrt{5+\sqrt{5+\sqrt{5+\sqrt{5+\cdots}}}}}\;

= 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 91]
Square root of i [93] \sqrt{i} \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 [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 92] Alladi–Grinstead constant [94] {\mathcal{A}_{AG}} e^{-1+\sum \limits_{k=2}^\infty \sum \limits_{n=1}^\infty \frac{1}{n k^{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}

[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 93] Sierpiński's constant [95] {K} \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)

= \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]]

[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 \frac {1}{\gamma} \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))

[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 94] Lebesgue constant (interpolation) [96][97] {L_1} \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{3 t}{2}}\rfloor}{\sin{\frac{t}{2}}}\, dt = \frac {1}{3} + \frac {2 \sqrt{3}}{\pi}
1/3 + 2*sqrt(3)/pi

T [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 95] Silver root
Tutte–Beraha constant [98]
\varsigma 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 [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 96] Euler Totient
constant
[99][100]
ET \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)

[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 [101] \sqrt[4]{5} \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 [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 97] Area of Ford circle [102] A_{CF} \sum_{q\ge 1} \sum_{ (p, q)=1 \atop 1 \le p < q }\pi \left( \frac{1}{2 q^2} \right)^2 \underset {\zeta() \text{= Riemann Zeta Function}} {= \frac{\pi}{4} \frac{\zeta(3)}{\zeta(4)} = \frac{45}{2} \frac{\zeta(3)}{\pi^3}}
pi Zeta(3)
/(4 Zeta(4))

[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 [Mw 98] Zeta(4) [103]

\zeta(4) \frac{\pi^4}{90} = \sum_{n=1}^\infty\frac{n^4} = \frac{1}{1^4} + \frac{1}{2^4} + \frac{1}{3^4} + \frac{1}{4^4} + \frac{1}{5^4} + ...
Sum[n=1 to ∞]
{1/n^4}

T [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.[104] {R_2} \frac{\sqrt{17}-1}{2} = \,\scriptstyle \sqrt{4+\sqrt{4+\sqrt{4+\sqrt{4+\sqrt{4+\sqrt{4+\cdots}}}}}} \,\, -1

= \,\scriptstyle \sqrt{4-\sqrt{4-\sqrt{4-\sqrt{4-\sqrt{4-\sqrt{4-\cdots}}}}}} \textstyle

(sqrt(17)-1)/2

A [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

{\pi} ^2 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
6 Sum[n=1 to ∞]
{1/n^2}

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 [Mw 99] Plastic number [105] {\rho} \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 [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 [106]

2\,ln\,\gamma \frac{\pi^2}{6ln(2)}
Pi^(2)/(6*ln(2))

T [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 [Mw 100] Regular paperfolding sequence [107][108] {P_f} \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]

[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 [Mw 101] Cubic recurrence constant [109] [110]

{\sigma_3} \prod_{n=1}^\infty 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}

[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 [Mw 102] Fractal dimension of the Koch snowflake [111]
{C_k} \frac{\log 4}{\log 3}
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[112]

2^{\,e} 2^e
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 [Mw 103] Meissel-Mertens constant [113] {M} \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 ? [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 [114] \gamma \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!}}
e^(π/2)

T [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 [115] \sqrt[3]{1} \begin{cases} \ \ 1 \\ -\frac{1}{2}+\frac{\sqrt{3}}{2}i \\ -\frac{1}{2}-\frac{\sqrt{3}}{2}i. \end{cases}
1,
E^(2i pi/3),
E^(-2i pi/3)

C A - [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 [Mw 104] Liouville number [116]

\text{£}_{Li} \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
Sum[n=1 to ∞]
{10^(-n!)}

T [1;9,1,999,10,9999999999999,1,9,999,1,9] 0.11000100000000000000000100...
0.06598 80358 45312 53707 [Mw 105] Lower limit of Tetration [117] {e}^{-e} \left(\frac {1}{e}\right)^e
1/(e^e)

[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[118] {\phi_{}}_3 \textstyle \frac{1+\sqrt[3]{19+3\sqrt{33}}+\sqrt[3]{19-3\sqrt{33}}}{3} = \scriptstyle \, 1+ \left(\sqrt[3]{\tfrac12 + \sqrt[3]{\tfrac12 + \sqrt[3]{\tfrac12 + ...}}}\right)^{-1}
(1/3)*(1+(19+3
*sqrt(33))^(1/3)
+(19-3
*sqrt(33))^(1/3))

A [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 [119] {ll_2} -\ln(\ln(2))
-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 [120]

\pi ^\pi \pi ^\pi
pi^pi

[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 [121] 2+\zeta(\tfrac12) {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)^(2n)
gamma_n)
/(2^n n!)}]

2-
[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 [Mw 106] Exponential reiterated constant [122] e^e \sum_{n=0}^\infty \frac{e^n}{n!} = \lim_{n \to \infty} \left(\frac {1+n}{n} \right)^{n^{-n}(1+n)^{1+n}}
Sum[n=0 to ∞]
{(e^n)/n!}

[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 [Mw 107] Masser–Gramain constant [123] {C} \gamma {\beta}(1) \! + \! {\beta}'(1) \! = \pi \! \left(-\!\ln \Gamma(\tfrac14)+\tfrac34 \pi+\tfrac12 \ln 2+\tfrac12 \gamma \right)

= \pi \! \left(-\!\ln (\tfrac14 !)+\tfrac34 \ln \pi -\tfrac32 \ln 2+\tfrac12 \, \gamma \right) \scriptstyle \gamma = \text{Euler–Mascheroni constant}= 0.5772156649\ldots \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]])

[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 [Mw 108] The ratio of a square and circle circumscribed [124] \frac{\pi}{2\sqrt 2} \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))
/(2n-1)}

T [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 [Mw 109] Backhouse's constant [125] {B} \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) = \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}})

[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 [Mw 110] Gibbs constant [126] {Si(\pi)}
Sin integral
\int_0^{\pi} \frac {\sin t}{t}\, dt = \sum \limits_{n=1}^\infty (-1)^{n-1} \frac{\pi^{2n-1}}{(2n-1)(2n-1)!}

= \pi- \frac{\pi^3}{3\cdot3!} + \frac{\pi^5}{5\cdot5!} - \frac{\pi^7}{7\cdot7!} + \cdots

SinIntegral[Pi]

[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 [Mw 111] Copeland–Erdős constant [127] {\mathcal{C}_{CE}} \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 [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 [Mw 112] Fractal dimension of the boundary of the dragon curve [128] {C_d} \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 [Mw 113] Grothendieck constant [129]

{K_{R}} \frac {\pi}{2 \log(1+\sqrt{2})}
pi/(2 log(1+sqrt(2)))

[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 [Mw 114] Hausdorff dimension, Sierpinski triangle [130] {log_2 3} \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^(2n+1) (2n+1))})/
(Sum[n=0 to ∞] {1/
(3^(2n+1) (2n+1))})

T [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 [Mw 115] Mills' constant [131] {\theta} \lfloor \theta^{3^{n}} \rfloor primes
Nest[ NextPrime[#^3] &, 2, 7]^(1/3^8)

[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 [Mw 116] Figure eight knot hyperbolic volume [132] {V_{8}} 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 =

\scriptstyle \frac{\sqrt{3}}\, \sum \limits_{n=0}^\infty \frac{(-1)^n}{27^n}\,\left\{\! \frac{(6n+1)^2} - \frac{(6n+2)^2} - \frac{(6n+3)^2} - \frac{(6n+4)^2} + \frac{(6n+5)^2}\!\right\}

6 integral[0 to pi/3]
{log(1/(2 sin (n)))}

[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 [Mw 117]
Hermite–Ramanujan constant[133] {R} e^{\pi\sqrt{163}}
e^(π sqrt(163))

T [262537412640768743;1,1333462407511,1,8,1,1,5,...] 1859 262537412640768743.999999999999250073
1.74540 56624 07346 86349 [Mw 118] Khinchin harmonic mean [134] {K_{-1}} \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}}

a1 ... an are elements of a continued fraction [a0; a1, a2, ..., an]

(log 2)/
(sum[n=1 to ∞]
{1/n log(1+
1/(n(n+2))}

[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 [Ow 5] Square root of the number e [135]

\sqrt {e} \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 [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 [Mw 119] Zeta(6) [136] \zeta(6) \frac{\pi^6}{945} \! = \! \prod_{n=1}^\infty \! \underset{p_n: \text{ prime}}{ \frac{1}} = \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 [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 [Mw 120] Trott constant [137] \mathrm{T}_1 \textstyle [1, 0, 8, 4, 1, 0, 1, 5, 1, 2, 2, 3, 1, 1, 1, 3, 6,...]

\tfrac 1{1+\tfrac 1{0+\tfrac 1{8+\tfrac 1{4+\tfrac 1{1+\tfrac 1{0+1{/\cdots}}}}}}}

[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 [Mw 121] Chaitin constant [138]
\Omega
\sum_{p \in P} 2^{-|p|} \overset {p: \text{ Halted program}}{ \underset{ P:\text{ Domain of all programs that stop.}} {\scriptstyle {|p|}:\text{Size in bits of program }p}}
T [0; 126, 1, 62, 5, 5, 3, 3, 21, 1, 4, 1] 1975 0.0078749969978123844
0.83462 68416 74073 18628 [Mw 122] Gauss constant [139] {G} \frac{1}{\mathrm{agm}(1, \sqrt{2})} = \frac{4 \sqrt{2} \,(\tfrac14 !)^2}{\pi ^{3/2}}= \frac{2}{\pi}\int_0^1\frac{dx}{\sqrt{1 - x^4}}
(4 sqrt(2)((1/4)!)^2)
/pi^(3/2)

T [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 [Mw 123] Ramanujan–Soldner constant[140][141] {\mu} \mathrm{li}(x) = \int\limits_0^x \frac{dt}{\ln t} = 0 {\color{White}{......}} li = Logarithmic integral

\mathrm{li}(x)\;=\;\mathrm{Ei}(\ln{x}) {\color{White}{........}} Ei = Exponential integral

FindRoot[li(x) = 0]

I [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 [Mw 124] Cahen's constant [142] \xi _{2} \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 kth term of Sylvester's sequence 2, 3, 7, 43, 1807, ...
Defined as: \, \, S_0= \, 2 , \,\, S_k= \, 1+\prod \limits_{n=0}^{k-1} S_n\text{ for}\;k>0

T [0; 1, 1, 1, 4, 9, 196, 16641, 639988804, ...] 1891 0.64341054628833802618225430775756476
1.41421 35623 73095 04880 [Mw 125] Square root of 2, Pythagoras constant.[143] \sqrt{2} \! \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)
/(2n-1)}

A [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 [Mw 126] Carlson–Levin constant [144] {\Gamma}(\tfrac12) \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
sqrt (pi)

T [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 [Ow 6] Musical interval between each half tone [145][146]

\sqrt[12]{2} \scriptstyle 440\, Hz. \textstyle 2^\frac{1}{12} \, 2^\frac{2}{12} \, 2^\frac{3}{12} \, 2^\frac{4}{12} \, 2^\frac{5}{12} \, 2^\frac{6}{12} \, 2^\frac{7}{12} \, 2^\frac{8}{12} \, 2^\frac{9}{12} \, 2^\frac{10}{12} \, 2^\frac{11}{12} \, 2

\scriptstyle {\color{white}...\color{black} Do_1\;\; Do\#\;\, Re\;\, Re\#\;\, Mi\;\; Fa\;\; Fa\#\; Sol\;\, Sol\#\, La\;\; La\#\;\; Si\;\, Do_2} \scriptstyle {\color{white}....\color{black}C_1\;\;\;\; C\#\;\;\;\, D\;\;\; D\#\;\;\, E\;\;\;\;\, F\;\;\;\, F\#\;\;\; G\;\;\;\; G\#\;\;\; A\;\;\;\, A\#\;\;\;\, B\;\;\; C_2}

2^(1/12)

A [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 [Mw 127] Gieseking constant [147] {\pi \ln \beta} \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)=

\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/((3n+2)^2)}
+Sum[n=1 to ∞]
{1/((3n+1)^2)})

[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 [Mw 128] Lemniscate constant [148] {\varpi} \pi \, {G} = 4 \sqrt{\tfrac2\pi}\,\Gamma{\left(\tfrac54 \right)^2} = \tfrac14 \sqrt{\tfrac{2}{\pi}}\,\Gamma {\left(\tfrac14 \right)^2} = 4 \sqrt{\tfrac2\pi}\left(\tfrac14 !\right)^2
4 sqrt(2/pi)
((1/4)!)^2

T [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 [Mw 129] Glaisher–Kinkelin constant [149]

{A} 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)}
e^(1/12-zeta´{-1})

T ? [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 [Mw 130] Digamma (1/4) [150] {\psi} (\tfrac14) -\gamma -\frac{\pi}{2} - 3\ln{2} = -\gamma+\sum_{n=0}^{\infty}\left(\frac{1}{n+1}-\frac{1}{n+\tfrac14}\right)
-EulerGamma
-\pi/2 -3 log 2

-[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 [Mw 131] Strongly Carefree constant [151]

K_{2} \prod_{n=1}^\infty \underset{p_n: \text{ prime}} {\left( 1-\frac{3 p_n-2}\right)} = \frac {6}{\pi ^2}\prod_{n=1}^\infty \underset{p_n: \text{ prime}} {\left( 1-\frac{1}\right)}
N[ prod[k=1 to ∞]
{1-(3*prime(k)-2)
/(prime(k)^3)}]

[0;3,2,19,3,12,1,5,1,5,1,5,2,1,1,1,1,1,3,7,...] 0.28674742843447873410789271278983845
1.78107 24179 90197 98523 [Mw 132] Exp.gamma,
Barnes G-function [152]
e^{\gamma} \prod_{n=1}^\infty \frac{e^{\frac{1}{n}}}{1+\tfrac1n} = \prod_{n=0}^\infty \left(\prod_{k=0}^n (k+1)^{(-1)^{k+1}{n \choose k}}\right)^{\frac{1}{n+1}} =

\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}

[1;1,3,1,1,3,5,4,1,1,2,2,1,7,9,1,16,1,1,1,2,...] 1.78107241799019798523650410310717954
3.62560 99082 21908 31193 [Mw 133] Gamma(1/4)[153] \Gamma(\tfrac14) 4 \left(\frac{1}{4}\right)! = \left(-\frac{3}{4}\right)!
4(1/4)!

T [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 [Mw 134] Somos' quadratic recurrence constant [154] {\sigma} \prod_{n=1}^\infty n^} = 1^{1/2} \; 2^{1/4} \; 3^{1/8} \cdots
prod[n=1 to ∞]
{n ^(1/2)^n}

T ? [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 [155] {\theta_m} \arctan \left(\sqrt{2}\right) = \arccos \left(\sqrt{\tfrac13}\right) \approx \textstyle {54.7356} ^{ \circ }
arctan(sqrt(2))

T [0;1,21,2,1,1,1,2,1,2,2,4,1,2,9,1,2,1,1,1,3,...] 0.95531661812450927816385710251575775
0.74759 79202 53411 43517 [Mw 135] Rényi's Parking Constant [156] {m} \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)]
* Int{n,0,∞}[ e^(- 2
*Gamma(0,n)) /n^2]

[0;1,2,1,25,3,1,2,1,1,12,1,2,1,1,3,1,2,1,43,...] 0.74759792025341143517873094383017817
1.44466 78610 09766 13365 [Mw 136] Steiner number, Iterated exponential Constant [157]
\sqrt[e]{e} e^{\frac{1}{e}}{\color{White}{...........}} = Upper Limit of Tetration
e^(1/e)

T [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 [Mw 137] Minimum value of función
ƒ(x) = xx [158]
{\left(\frac{1}{e}\right)}^\frac{1}{e} {e}^{-\frac{1}{e}} {\color{White}{..........}} = Inverse Steiner Number
e^(-1/e)

[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 [Mw 138] Pólya Random walk constant [159] {p(3)} 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\sqrt{\tfrac23}\;\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])

[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 [Mw 139] Bloch–Landau constant [160] {L} = \frac {\Gamma(\tfrac13)\;\Gamma(\tfrac{5}{6})} {\Gamma(\tfrac{1}{6})} = \frac {(-\tfrac23)!\;(-1+\tfrac56)!} {(-1+\tfrac16)!}
gamma(1/3)
*gamma(5/6)
/gamma(1/6)

[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 [Mw 140] [Ow 7] MRB Constant, Marvin Ray Burns [161][162][163] C_ \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)}

[0;5,3,10,1,1,4,1,1,1,1,9,1,1,12,2,17,2,2,1,...] 1999 0.18785964246206712024851793405427323
1.27323 95447 35162 68615 Ramanujan–Forsyth series[164] \frac {4}{\pi} \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 ∞]
\qquad \scriptstyle x \in (3.8284;\, 3.8495)


T [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 [Mw 143] Feigenbaum constant α[167] \alpha \lim_{n \to \infty}\frac {d_n}{d_{n+1}} T ? [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 [Mw 144] Golomb–Dickman constant [168]

{\lambda} \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]

[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 145] Gelfond constant [169]

{e}^{\pi} (-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 [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 [170] e^2 \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 [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 146] Hafner–Sarnak–McCurley constant (1) [171] {\sigma} \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}

[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 147] Hafner–Sarnak–McCurley constant (2) [172] \frac{1}{\zeta(2)} \frac{6}{\pi^2} = \prod_{n = 0}^\infty \underset{p_n: \text{ prime}}{\! \left(\! 1- \frac{1}^{10^n-1}\frac{k}{10^{kn-9\sum_{j=0}^{n-1}10^j(n-j-1)}} T [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 149] Landau-Ramanujan constant [174]

K \frac1{\sqrt2}\prod_{p\equiv3\!\!\!\!\!\mod \! 4}\!\! \underset{\!\!\!\!\!\!\!\! p: \text{ prime}}{\left(1-\frac1{p^2}\right)^{-\frac{1}{2}}}\!\!=\frac\pi4\prod_{p\equiv1\!\!\!\!\!\mod \!4}\!\! \underset{\!\!\!\! p: \text{ prime}}{\left(1-\frac1{p^2}\right)^\frac{1}{2}} T ? [0;1,3,4,6,1,15,1,2,2,3,1,23,3,1,1,3,1,1,6,4,...] 0.76422365358922066299069873125009232
1.92878 00... [Mw 150] Wright constant [175]

{\omega} \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 [1; 1, 13, 24, 2, 1, 1, 3, 1, 1, 3] 1.9287800...
2.71828 18284 59045 23536 [Mw 151] Number e, Euler's number [176] {e} \! \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 [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 152] Inverse of Number e [177]

\frac{1}{e} \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 [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 [178] {H}_{2n+1} \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^}}}}
2^-3^-4^-5^-6^
-7^-8^-9^-10^
-11^-12^-13 …

[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 [179] {H}_{2n} \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^}}}}
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 153] π number, Archimedes number [180] \pi \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 [3;7,15,1,292,1,1,1,2,1,3,1,14,2,1,1,2,2,2,...] 3.14159265358979323846264338327950288
0.46364 76090 00806 11621 Machin–Gregory series[181] \arctan \frac {1}{2} \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 [0;2,6,2,1,1,1,6,1,2,1,1,2,10,1,2,1,2,1,1,1,...] 0.46364760900080611621425623146121440
1.90216 05831 04 [Mw 154] Brun 2 constant = Σ inverse of Twin primes [182] {B}_{\,2} \textstyle \underset{ p,\, p+2: \text{ prime}}{\sum(\frac1{p}+\frac1{p+2})} = (\frac1{3} \! + \! \frac1{5}) + (\tfrac1{5} \! + \! \tfrac1{7}) + (\tfrac1{11} \! + \! \tfrac1{13}) + \cdots [1; 1, 9, 4, 1, 1, 8, 3, 4, 4, 2, 2] 1.902160583104
0.87058 83799 75 [Mw 155] Brun 4 constant = Σ inv.prime quadruplets [183]

{B}_{\,4} \textstyle {\sum(\frac1{p}+\frac1{p+2}+\frac1{p+4}+\frac1{p+6})} \scriptstyle \quad {p,\; p+2,\; p+4,\; p+6: \text{ prime}}

\textstyle{\left(\tfrac1{5} + \tfrac1{7} + \tfrac1{11} + \tfrac1{13}\right)}+ \left(\tfrac1{11} + \tfrac1{13} + \tfrac1{17} + \tfrac1{19}\right)+ \dots

[0; 1, 6, 1, 2, 1, 2, 956, 3, 1, 1] 0.870588379975

0.63661 97723 67581 34307 [Mw 156] [Ow 8]

Buffon constant[184] \frac{2}{\pi} \frac{\sqrt2}2 \cdot \frac{\sqrt{2+\sqrt2}}2 \cdot \frac{\sqrt{2+\sqrt{2+\sqrt2}}}2 \cdots Viète product
2/Pi

T [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 157] Euler–Gompertz constant [185] {G} \! \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 [0;1,1,2,10,1,1,4,2,2,13,2,4,1,32,4,8,1,1,1,...] 0.59634736232319407434107849936927937
i ··· [Mw 158]
sqrt(-1)

C I 1501
to
1576
i
0.69777 46579 64007 98200 [Mw 159] Continued fraction constant, Bessel function[187] {C}_{CF} \frac{I_1(2)}{I_0(2)} = \frac{ \sum \limits_{n = 0}^\infty \frac{n}{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 [0;1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,...]
= [0;p+1], p∈ℕ
0.69777465796400798200679059255175260
2.74723 82749 32304 33305 Ramanujan nested radical [188]

R_{5} \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 160] Omega constant, Lambert W function [189] {\Omega} \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^}}}}
Sum[n=1 to ∞]
{(-n)^(n-1)/n!}

T [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) [190] {\beta} (3) \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 [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 [191] \sqrt{5} \scriptstyle (n = 5) \displaystyle \sum_{k=0}^{n-1} e^{\frac{2 k^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 [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 161] Prévost constant Reciprocal Fibonacci constant[192] \Psi \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
Sum[n=1 to ∞]
{1/Fibonacci[n]}

I [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 162] Khinchin's constant [193] K_{\,0} \prod_{n=1}^\infty \left\right]^{\ln n/\ln 2}
Prod[n=1 to ∞]
{(1+1/(n(n+2)))
^(ln(n)/ln(2))}

T [2;1,2,5,1,1,2,1,1,3,10,2,1,3,2,24,1,3,2,...] 1934 2.68545200106530644530971483548179569