ਗਣਿਤਿਕ ਸਥਿਰਾਂਕ ਅਤੇ ਫੰਕਸ਼ਨ

ਇੱਕ ਗਣਿਤਿਕ ਕੌਂਸਟੈਂਟ ਜਾਂ ਸਥਿਰਾਂਕ ਇੱਕ ਅਜਿਹਾ ਨੰਬਰ (ਸੰਖਿਆ) ਹੁੰਦਾ ਹੈ, ਜਿਸਦਾ ਕੈਲਕੁਲੇਸ਼ਨਾਂ ਲਈ ਇੱਕ ਖਾਸ ਅਰਥ ਹੁੰਦਾ ਹੈ। ਉਦਾਹਰਨ ਦੇ ਤੌਰ 'ਤੇ, ਸਥਿਰਾਂਕ π (ਪਾਈ) ਦਾ ਅਰਥ ਹੈ ਕਿਸੇ ਚੱਕਰ ਦੇ ਘੇਰੇ ਦੀ ਲੰਬਾਈ ਦਾ ਇਸਦੇ ਡਾਇਆਮੀਟਰ (ਅਰਧ-ਵਿਆਸ) ਪ੍ਰਤਿ ਅਨੁਪਾਤ (ਰੇਸ਼ੋ)। ਇਹ ਮੁੱਲ ਹਮੇਸ਼ਾ ਹੀ ਕਿਸੇ ਚੱਕਰ ਲਈ ਇਹੀ ਰਹਿੰਦਾ ਹੈ।

## ਸਾਰਣੀਆਂ ਦੀ ਬਣਤਰ 
*ਸਥਿਰਾਂਕ ਦਾ ਸੰਖਿਅਕ **ਮੁੱਲ** ਅਤੇ ਗਣਿਤ-ਸੰਸਾਰ ਨਾਲ ਜਾਂ OEIS ਵਿਕੀ ਨਾਲ ਲਿੰਕ। 
***LaTeX**: ਫੌਰਮੈਟ ਵਿੱਚ ਫਾਰਮੂਲਾ ਜਾਂ ਸੀਰੀਜ਼। 
***ਫਾਰਮੂਲਾ**: ਵੌਲਫ੍ਰਾਮ ਅਲਫਾ ਪ੍ਰੋਗਰਾਮ ਅੰਦਰ ਵਰਤੋਂ ਵਾਸਤੇ। 
***OEIS**: On-Line Encyclopedia of।nteger Sequences। 
***ਨਿਰੰਤਰ ਭਿੰਨ**: ਰੂਪ [to integer; frac1, frac2, frac3, ...] ਵਿੱਚ ਸਰਲ, ਜੇਕਰ ਪੀਰੀਔਡਿਕ ਹੋਵੇ ਤਾਂ । 
***ਸਾਲ**: ਸਥਿਰਾਂਕ ਦੀ ਖੋਜ, ਜਾਂ ਵਿਦਵਾਨ ਦੀਆਂ ਤਰੀਕਾਂ। 
***ਵੈਬ ਫੌਰਮੈਟ**: ਵੈਬ ਬਰਾਊਜ਼ਰਾਂ ਲਈ ਢੁਕਵੇਂ ਫੌਰਮੈਟਾਂ ਵਿੱਚ ਮੁੱਲ। 
***Nº**: ਨੰਬਰ ਕਿਸਮਾਂ। 
** R – ਰੈਸ਼ਨਲ ਸੰਖਿਆ
**। – ਗੈਰ-ਰੈਸ਼ਨਲ ਸੰਖਿਆ
** A – ਅਲਜਬ੍ਰਿਕ ਸੰਖਿਆ
** T – ਟ੍ਰਾਂਸਡੈਂਟਲ ਸੰਖਿਆ
** C – ਗੈਰ-ਵਾਸਤਵਿਕ ਕੰਪਲੈਕਸ ਨੰਬਰ

## ਸਥਿਰਾਂਕਾਂ ਅਤੇ ਫੰਕਸ਼ਨਾਂ ਦੀ ਸਾਰਣੀ 
*ਨਾਮ, ਮੁੱਲ, OEIS ਅਦਿ ਉੱਤੇ ਕਲਿੱਕ ਕਰਕੇ ਤੁਸੀਂ ਸੂਚੀ ਦੇ ਔਰਡਰ ਨੂੰਚੁਣ ਸਕਦੇ ਹੋ..*

 {\sin(\frac{t}{2})} \,d t 
||
```mathematica
1/5 + sqrt(25 - 
 2*sqrt(5))/Pi
```

|style="text-align:center;"|***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
||Vardi constant
||
| bgcolor=#e0f0f0 align=center|{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 
|-

| ± 
||Area of the Mandelbrot fractal
|bgcolor=#ffffff align=center|
| bgcolor=#e0f0f0 align=center|\gamma 
|| This is conjectured to be: \sqrt{6\pi -1} - e = 1.506591651\cdots  
||
||
||
||[1;1,1,37,2,2,1,10,1,1,2,2,4,1,1,1,1,5,4,...]
||1912
|| 1.50659177 +/- 0.00000008 
|-

|1.61111 49258 08376 736 111···111  27224 36828  *183213 ones*
|| Exponential factorial constant 
||
| bgcolor=#e0f0f0 align=center|{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
||
|style="text-align:center;"|***T***
||
||[1; 1, 1, 1, 1, 2, 1, 808, 2, 1, 2, 1, 14,...]
||
|| 1.61111492580837673611111111111111111 
|-

|1.11786 41511 89944 97314
||Goh-Schmutz constant
||
| bgcolor=#e0f0f0 align=center| C_{GS} 
|| \int^\infty_0\frac{\log(s+1)}{e^s-1} \ ds =
\! - \! \sum_{n=1}^\infty \frac {e^n}{n} Ei(-n)

* : Exponential।ntegral
||
```mathematica
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***
||Fixed points Super-Logarithm Tetration
||
| bgcolor=#e0f0f0 align=center| {-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  different to  0, 1, e, e^e, e^{e^e}
, etc.
||`-W(-1)`
where W=ProductLog
Lambert W function 
|style="text-align:center;"|***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
||Bernstein's constant
||
| bgcolor=#e0f0f0 align=center|{\beta}
|| \approx \frac {1}{2\sqrt {\pi}}
||
```mathematica
1/(2 sqrt(pi))
```

|style="text-align:center;"|***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
||Twin Primes Constant
||
| bgcolor=#e0f0f0 align=center|{C}_{2}
||\prod_{p=3}^\infty \frac{p(p-2)}{(p-1)^2} 
||
```mathematica
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
||Fibonacci Factorial constant
||
| bgcolor=#e0f0f0 align=center|F 
||  \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)
||
```mathematica
prod[n=1 to ∞] 
 {1-((sqrt(5) -3)/2)^n}
```

||
||
||[1;4,2,2,3,2,15,9,1,2,1,2,15,7,6,21,3,5,1,23,...]
||
|| 1.22674201072035324441763023045536165 
|-

|0.11494 20448 53296 20070
||Kepler–Bouwkamp constant
|bgcolor=#ffffff align=center|
| bgcolor=#e0f0f0 align=center|{\rho}
|| \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) ...
||
```mathematica
prod[n=3 to ∞]
{cos(pi/n)}
```

||
||
||[0;8,1,2,2,1,272,2,1,41,6,1,3,1,1,26,4,1,1,...]
||
|| 0.11494204485329620070104015746959874 
|-

|1.78723 16501 82965 93301
||Komornik–Loreti constant
||
| bgcolor=#e0f0f0 align=center|{q}
||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
||
```mathematica
FindRoot[(prod[n=0 to ∞] 
{1-1/(x^2^n)}+(x-2)
/(x-1))= 0, {x, 1.7}, 
WorkingPrecision->30]
```

|style="text-align:center;"|***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
||Bronze ratio
||
| bgcolor=#e0f0f0 align=center|{\sigma}_{\,Rr} 
||\frac {3+\sqrt{13}}{2} = 1+\sqrt{3+\sqrt{3+\sqrt{3+\sqrt{3+\cdots}}}} 
||
```mathematica
(3+sqrt 13)/2
```

|style="text-align:center;"|***A***
||
||[3;3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,...]  = [3;,...]
||
|| 3.30277563773199464655961063373524797 
|-

|0.82699 33431 32688 07426
||Disk Covering
|bgcolor=#ffffff align=center|
| bgcolor=#e0f0f0 align=center|{C_5}
||{\frac{1}{{\sum\limits_{n=0}^\infty \frac{1}{\binom{3n+2}{2}}}}}=\frac{3\sqrt3}{2\pi}
||
```mathematica
3 Sqrt[3]/(2 Pi)
```

|style="text-align:center;"|***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
||Gelfond–Schneider constant
||
| bgcolor=#e0f0f0 align=center| G_{\,GS} 
||2^{\sqrt{2}}
||
```mathematica
2^sqrt{2}
```

|style="text-align:center;"|***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
||Khinchin-Lévy constant
||
| bgcolor=#e0f0f0 align=center|\gamma 
||  e^{\pi^2/(12\ln2)}
||
```mathematica
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
||Chi Function  Hyperbolic cosine integral
|bgcolor=#ffffff align=center|
| bgcolor=#e0f0f0 align=center|{\operatorname{Chi()}}
||\gamma + \int_0^x\frac{\cosh t-1}{t}\,dt 
 \scriptstyle \gamma \, \text{= Euler–Mascheroni constant= 0.5772156649...}
||
```mathematica
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
||Viswanath constant
||
| bgcolor=#e0f0f0 align=center|{C}_{Vi}
||\lim_{n \to \infty}|a_n|^\frac{1}{n} where *an* = Fibonacci sequence
||
```mathematica
lim_(n->∞) 
|a_n|^(1/n)
```
 
|style="text-align:center;"|***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
||Favard constant
||
| bgcolor=#e0f0f0 align=center|\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 
||
```mathematica
sum[n=1 to ∞]
{1/((2n-1)^2)}
```

|style="text-align:center;"|***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
|| 
| bgcolor=#e0f0f0 align=center|\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*
||
```mathematica
sqrt (2 pi)
```

|style="text-align:center;"|***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
||
| bgcolor=#e0f0f0 align=center| \sqrt{\tau e}
|| \sqrt{2 \pi e}
||
```mathematica
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
||Lochs constant
||
| bgcolor=#e0f0f0 align=center|{\text{£}_{_{Lo}}}
|| \frac {6 \ln 2 \ln 10}{ \pi^2} 
||
```mathematica
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
||Area bounded by the eccentric rotation of Reuleaux triangle 
|bgcolor=#ffffff align=center|
| bgcolor=#e0f0f0 align=center|\mathcal{T}_R
|| a^2 \cdot \left(2\sqrt{3} + {\frac{\pi}{6}} - 3 \right) where *a*= side length of the square
||
```mathematica
2 sqrt(3)+pi/6-3
```

|style="text-align:center;"|***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
|
|bgcolor=#e0f0f0 align=center|\mathcal{C}_2
|| \underset{ p_n: \, {prime}}{\prod_{n = 1}^\infty \left(1 - \frac{1}{p_n(p_n+1)}\right)} 
||
```mathematica
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
||Connective constant
|bgcolor=#ffffff align=center|
| bgcolor=#e0f0f0 align=center|{\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
||
```mathematica
sqrt(2+sqrt(2))
```

|style="text-align:center;"|***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
||Gauss-Kuzmin-Wirsing constant
||
| bgcolor=#e0f0f0 align=center|{\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
||Favard constant K1  Wallis product
||
| bgcolor=#e0f0f0 align=center|{\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 
||
```mathematica
Prod[n=1 to ∞] 
 {(4n^2)/(4n^2-1)}
```

|style="text-align:center;"|***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
||Erdős–Borwein constant
||
| bgcolor=#e0f0f0 align=center|{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} \! + \! ...
||
```mathematica
sum[n=1 to ∞]
{1/(2^n-1)}
```

|style="text-align:center;"|***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
||Phi, Golden ratio
||
| bgcolor=#e0f0f0 align=center|{\varphi}
|| \frac{1 + \sqrt{5}}{2} = \sqrt{1 + \sqrt{1 + \sqrt{1 + \sqrt{1 + \cdots}}}} 
||
```mathematica
(1+5^(1/2))/2
```

|style="text-align:center;"|***A***
||
||[0;1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,...] = [0;,...]
||-300 ~
|| 1.61803398874989484820458683436563811 
|-

|1.64493 40668 48226 43647
||Riemann Function Zeta(2)
||
| bgcolor=#e0f0f0 align=center|{\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
||
```mathematica
Sum[n=1 to ∞]
{1/n^2}
```

|style="text-align:center;"|***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
||Theodorus constant
||
| bgcolor=#e0f0f0 align=center|\sqrt{3} 
|| \sqrt[3]{3 \,\sqrt[3]{3 \, \sqrt[3]{3 \,\sqrt[3]{3 \,\sqrt[3]{3 \,\cdots}}}}} 
||
```mathematica
(3(3(3(3(3(3(3) 
 ^1/3)^1/3)^1/3) 
 ^1/3)^1/3)^1/3) 
 ^1/3 ...
```

|style="text-align:center;"|***A***
||
||[1;1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,...]  = [1;,...]
||-465  to  -398
|| 1.73205080756887729352744634150587237 
|-

|1.75793 27566 18004 53270
||Kasner number
||
| bgcolor=#e0f0f0 align=center|{R}
||\sqrt{1 + \sqrt{2 + \sqrt{3 + \sqrt{4 + \cdots}}}} 
|| 
```mathematica
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
||Universal parabolic constant
||
| bgcolor=#e0f0f0 align=center| {P}_{\,2} 
||\ln(1 + \sqrt2) + \sqrt2 \; = \; \operatorname{arcsinh}(1)+\sqrt{2}
||
```mathematica
ln(1+sqrt 2)+sqrt 2
```

|style="text-align:center;"|***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
||Silverman constant
||
|bgcolor=#e0f0f0 align=center|{\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, *σ*1() = Divisor function.
||
```mathematica
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
||Area of the regular hexagon with side equal to 1
|bgcolor=#ffffff align=center|
 |bgcolor=#e0f0f0 align=center|\mathcal{A}_6
|| \frac{3 \sqrt{3}}{2} 
||
```mathematica
3 sqrt(3)/2
```

|style="text-align:center;"|***A***
||
||[2;1,1,2,20,2,1,1,4,1,1,2,20,2,1,1,4,1,1,2,20,...]  [2;]
||
|| 2.59807621135331594029116951225880855 
|-

|0.66131 70494 69622 33528
|| Feller-Tornier constant
||
|bgcolor=#e0f0f0 align=center|{\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}
||
```mathematica
[prod[n=1 to ∞] 
 {1-2/prime(n)^2}] 
 /2 + 1/2
```

|style="text-align:center;"|***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
||Baxter's  Four-coloring  constant
|bgcolor=#ffffff align=center| Mapamundi  Four-Coloring
 |bgcolor=#e0f0f0 align=center|\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 
||
```mathematica
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
||Tetranacci constant

||
|bgcolor=#e0f0f0 align=center|\mathcal{T}
||Positive root of : \;\; x^4-x^3-x^2-x-1=0
||
```mathematica
Root[x+x^-4-2=0]
```

|style="text-align:center;"|***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
||DeVicci's tesseract constant

||
|bgcolor=#e0f0f0 align=center|{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
||
```mathematica
Root[4*x^8-28*x^6 
 -7*x^4+16*x^2+16 
 =0]
```

|style="text-align:center;"|***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
||Niven's constant
||
| bgcolor=#e0f0f0 align=center|{C}
||1+\sum_{n = 2}^\infty \left(1-\frac{1}{\zeta(n)} \right)
||
```mathematica
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
||Relationship among the area of an equilateral triangle and the inscribed circle.
||
| bgcolor=#e0f0f0 align=center| \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 
||
```mathematica
Sum[1/(n 
Binomial[2 n, n])
, {n, 1, ∞}]
```

|style="text-align:center;"|***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
|| Hermite constant
||
| bgcolor=#e0f0f0 align=center| \gamma_{_{2}} 
|| \frac{2}{\sqrt{3}} = \frac{1}{\cos \, (\frac{\pi}{6})} 
||
```mathematica
2/sqrt(3)
```

|style="text-align:center;"|***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;]
|| 
|| 1.15470053837925152901829756100391491 
|-

|0.41245 40336 40107 59778
||Prouhet–Thue–Morse constant
||
| bgcolor=#e0f0f0 align=center| \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})
||
|style="text-align:center;"|***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
||Pell constant
||
| bgcolor=#e0f0f0 align=center|{\mathcal{P}_{_{Pell}}}
||1- \prod_{n = 0}^\infty \left(1-\frac{1}{2^{2n+1}}\right) 
||
```mathematica
N[1-prod[n=0 to ∞] 
 {1-1/(2^(2n+1)}]
```

|style="text-align:center;"|***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
||Laplace limit
||
| bgcolor=#e0f0f0 align=center|{\lambda}
|| \frac{ x \; e^\sqrt{x^2+1}}{\sqrt{x^2+1}+1} = 1
||
```mathematica
(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
||Hall-Montgomery Constant
||
| bgcolor=#e0f0f0 align=center| {{\delta}_{_{0}}} 
||  1 + \frac{\pi^2}{6} +2 \; \mathrm{Li}_2 \left(-\sqrt{e}\;\right) 
\quad \mathrm{Li}_2 \, \scriptstyle \text{= Dilogarithm integral} 
||
```mathematica
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
||Calabi triangle constant
||
| bgcolor=#e0f0f0 align=center| {C_{_{CR}}} 
||  {1 \over 3} + {(-23 + 3i \sqrt{237})^{\tfrac13} \over 3 \cdot 2^{\tfrac23}} + {11 \over 3 (2 (-23 + 3i \sqrt{237}))^{\tfrac13}} 
||
```mathematica
FindRoot[ 
 2x^3-2x^2-3x+2 
 ==0, {x, 1.5}, 
 WorkingPrecision->40]
```

|style="text-align:center;"|***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
||Gamma(3/4)
||
| bgcolor=#e0f0f0 align=center|\Gamma(\tfrac34)
||\left(-1+\frac{3}{4}\right)! = \left(-\frac{1}{4}\right)!
||
```mathematica
(-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
||Apéry's constant
||
| bgcolor=#e0f0f0 align=center|\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} 
||
```mathematica
Sum[n=1 to ∞]
{1/n^3}
```

|style="text-align:center;"|***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
||Catalan's constant
||
| bgcolor=#e0f0f0 align=center|{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} 
||
```mathematica
Sum[n=0 to ∞]
{(-1)^n/(2n+1)^2}
```

|style="text-align:center;"|***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
||Beta(1)
||
| bgcolor=#e0f0f0 align=center|{\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
||
```mathematica
Sum[n=0 to ∞]
{(-1)^n/(2n+1)}
```

|style="text-align:center;"|***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
||Heath-Brown–Moroz constant
||
| bgcolor=#e0f0f0 align=center|{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)} 
||
```mathematica
N[prod[n=1 to ∞] 
 {((1-1/prime(n))^7) 
 *(1+(7*prime(n)+1) 
 /(prime(n)^2))}]
```

|style="text-align:center;"|***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 ।nfinite  Tetration of *i*
||
| bgcolor=#e0f0f0 align=center||{}^\infty {i} | 
|| \lim_{n \to \infty} \left | {}^n i \right | =\left | \lim_{n \to \infty} \underbrace{i^{i^{\cdot^{\cdot^{i}}}}}_n \right |
||
```mathematica
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
||Sophomore's dream1  J.Bernoulli
||
| bgcolor=#e0f0f0 align=center|{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} 
||
```mathematica
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
||Sophomore's dream2  J.Bernoulli
||
| bgcolor=#e0f0f0 align=center|{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
||
```mathematica
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
||Primorial constant   Sum of the product of inverse of primes 
||
| bgcolor=#e0f0f0 align=center|{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}} 
||
```mathematica
Sum[k=1 to ∞] 
 (prod[n=1 to k] 
 {1/prime(n)})
```

|style="text-align:center;"|***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
||Plouffe's gamma constant
||
| bgcolor=#e0f0f0 align=center|
|| \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)
||
```mathematica
Arctan(1/2)/pi
```

|style="text-align:center;"|***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
||Plouffe's A constant
||
| bgcolor=#e0f0f0 align=center|{A}
|| \frac{1}{2 \pi} 
||
```mathematica
1/(2 pi)
```

|style="text-align:center;"|***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
||Dimer constant 2D,  Domino tiling
||
| bgcolor=#e0f0f0 align=center|{\frac{C}{\pi}}
C=Catalan
|| \int\limits_{-\pi}^{\pi} \frac{\cosh^{-1}\left(\frac{\sqrt{\cos(t)+3}}{\sqrt2}\right)}{4\pi}\,dt 
||
```mathematica
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*)
||
| bgcolor=#e0f0f0 align=center|{i}\,!
|| \Gamma (1+i) = i \, \Gamma (i) = \int\limits_0^\infty \frac{t^i}{e^t} \mathrm{d} t
||
```mathematica
Integral_0^∞ 
 t^i/e^t dt
```

|style="text-align:center;"|***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
||Wallis Constant
||
| bgcolor=#e0f0f0 align=center| W 
|| \sqrt[3]{\frac{45-\sqrt{1929}}{18}}+\sqrt[3]{\frac{45+\sqrt{1929}}{18}}
||
```mathematica
(((45-sqrt(1929)) 
 /18))^(1/3)+ 
 (((45+sqrt(1929)) 
 /18))^(1/3)
```

|style="text-align:center;"|***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
||Sarnak constant
||
| bgcolor=#e0f0f0 align=center|{C_{sa} }
|| \prod_{p>2} \Big(1 - \frac{p+2}{p^3}\Big) 
||
```mathematica
N[prod[k=2 to ∞] 
 {1-(prime(k)+2) 
 /(prime(k)^3)}]
```

|style="text-align:center;"|***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
||Time constant
||
| bgcolor=#e0f0f0 align=center|{\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 
||
```mathematica
lim_(n->∞) (1- !n/n!) 
 !n=subfactorial
```

|style="text-align:center;"|***T***
||
||[0;1,1,1,2,1,1,4,1,1,6,1,1,8,1,1,10,1,1,12,1,...]  = [0;1,], n∈ℕ 
||
|| 0.63212055882855767840447622983853913 
|-

|1.04633 50667 70503 18098
||Minkowski-Siegel mass constant 
||
| bgcolor=#e0f0f0 align=center| F_1 
|| \prod_{n=1}^{\infty} \frac{n!}{\sqrt{2\pi n}\left(\frac{n}{e}\right)^n \sqrt[12]{1+\tfrac1{n}}}
||
```mathematica
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
||Lemniscate Constant
||
| bgcolor=#e0f0f0 align=center|2\varpi
||\frac{[\Gamma(\tfrac14)]^2}{\sqrt{2 \pi}} = 
4\int^1_0 \frac{dx}{\sqrt{(1-x^2)(2-x^2)}} 
||
```mathematica
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
||Robbins constant
||
|bgcolor=#e0f0f0 align=center|\Delta(3)
|| \frac{4 \! + \! 17\sqrt2 \! -6 \sqrt3 \! -7\pi}{105} \! + \! \frac{\ln(1 \! + \! \sqrt2)}{5} \! + \! \frac{2\ln(2 \! + \! \sqrt3)}{5} 
||
```mathematica
(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 
|-

|1.30357 72690 34296 39125
||Conway constant
||
| bgcolor=#e0f0f0 align=center|{\lambda}
|| \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}
||
|style="text-align:center;"|***A***
||
||[1;3,3,2,2,54,5,2,1,16,1,30,1,1,1,2,2,1,14,1,...]
||1987
|| 1.30357726903429639125709911215255189 
|-

|1.18656 91104 15625 45282
||Khinchin–Lévy constant
||
| bgcolor=#e0f0f0 align=center|{\beta}
||\frac {\pi^2}{12\,\ln 2}
||
```mathematica
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
||
|bgcolor=#e0f0f0 align=center|\beta_3
||\int^1_0 \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)
||
```mathematica
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
||Kempner Serie(0)
||
| bgcolor=#e0f0f0 align=center|{K_0}
||1{+}\frac12{+}\frac13{+}\cdots{+}\frac19{+}\frac1{11}{+}\cdots{+}\frac1{19}{+}\frac1{21}{+}\cdots
{+}\frac1{99}{+}\frac1{111}{+}\cdots{+}\frac1{119}{+}\frac1{121}{+}\cdots

(Excluding all denominators containing 0.)
||
```mathematica
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
||Lebesgue constant 
||
| bgcolor=#e0f0f0 align=center|{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)
||
```mathematica
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
||2nd du Bois-Reymond constant
||
| bgcolor=#e0f0f0 align=center|{C_2}
||\frac{e^2-7}{2} = \int_0^\infty \left|{\frac{d}{dt}\left(\frac{\sin t}{t}\right)^n}\right|\,dt-1 
||
```mathematica
(e^2-7)/2
```
 
|style="text-align:center;"|***T***
||
||[0;5,7,9,11,13,15,17,19,21,23,25,27,29,31,...] = [0;], p∈ℕ
||
|| 0.19452804946532511361521373028750390 
|-

|0.78853 05659 11508 96106
||Lüroth constant
||
| bgcolor=#e0f0f0 align=center|C_L
||\sum_{n = 2}^\infty \frac{\ln\left(\frac{n}{n-1}\right)}{n}
||
```mathematica
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
||Foias constant α
||
|bgcolor=#e0f0f0 align=center|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 *x*1 = *&alpha;* then the sequence diverges to ∞. When *x*1 = *&alpha;*, \, \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
||Foias constant β 
||
|bgcolor=#e0f0f0 align=center|F_\beta
|| x^{x+1} = (x+1)^x
||
```mathematica
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
||Nielsen-Ramanujan constant
||
| bgcolor=#e0f0f0 align=center|\frac{{\zeta}(2)}{2}
|| \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 
||
```mathematica
Sum[n=1 to ∞]
{((-1)^(n+1))/n^2}
```

|style="text-align:center;"|***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
||Natural logarithm of 2
||
| bgcolor=#e0f0f0 align=center|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} 
||
```mathematica
Sum[n=1 to ∞]
{(-1)^(n+1)/n}
```

|style="text-align:center;"|***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
||Weierstrass constant
||
| bgcolor=#e0f0f0 align=center|\sigma(\tfrac12)
|| \frac{e^{\frac{\pi}{8}}\sqrt{\pi}}{4 \cdot 2^{3/4} {(\frac {1}{4}!)^2}}
||
```mathematica
(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
||Euler–Mascheroni constant
||
| bgcolor=#e0f0f0 align=center|{\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)
||
```mathematica
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
||
|bgcolor=#e0f0f0 align=center|\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}}
|| 
```mathematica
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
||Golden Spiral 
||
| bgcolor=#e0f0f0 align=center| c 
|| \varphi ^ \frac{2}{\pi} = \left(\frac{1 + \sqrt{5}}{2}\right)^{\frac{2}{\pi}}
||
```mathematica
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
||Stephens constant
||
| bgcolor=#e0f0f0 align=center| C_S 
|| \prod_{n = 1}^\infty \left(1 - \frac{p}{p^3-1}\right) 
||
```mathematica
Prod[n=1 to ∞] 
 {1-hprime(n) 
 /(hprime(n)^3-1)}
```

|style="text-align:center;"|***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
||Dottie number
||
|bgcolor=#e0f0f0 align=center|d
|| \lim_{x\to \infty} \cos^{[x]}(c) = \lim_{x\to \infty} \underbrace{\cos(\cos(\cos(\cdots(\cos(c)))))}_x
||
```mathematica
cos(c)=c
```

|style="text-align:center;"|***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
||Taniguchi constant
||
|bgcolor=#e0f0f0 align=center| C_T 
|| \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) 
\scriptstyle p_{n}= \, \text{prime} 
||
```mathematica
Prod[n=1 to ∞] {1 
 -3/ithprime(n)^3 
 +2/ithprime(n)^4 
 +1/ithprime(n)^5 
 -1/ithprime(n)^6}
```

|style="text-align:center;"|***T*** ?
|
||[0;1,2,9,3,1,2,9,11,1,13,2,15,1,1,1,2,4,1,1,1,...]
||?
|| 0.67823449191739197803553827948289481 
|-

|1.85407 46773 01371 91843
||Gauss' Lemniscate constant
||
|bgcolor=#e0f0f0 align=center| L \text{/}\sqrt{2}
||\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}}
\scriptstyle \Gamma() \text{= Gamma function} 
||
```mathematica
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
||
|bgcolor=#e0f0f0 align=center| Pr_1
|| \prod_{n = 2}^\infty \Big(1 + \frac{1}{n}\Big)^\frac{1}{n}
||
```mathematica
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 
||
|bgcolor=#e0f0f0 align=center| \partial 
|| \sum_{n=1}^{\infty} \frac{1}{\sqrt{n^3} + \sqrt{n}} =
\sum_{n=1}^{\infty} \frac{1}{\sqrt{n} (n+1)}
||
```mathematica
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 
||
|bgcolor=#e0f0f0 align=center| 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}}}}}
||
```mathematica
(sqrt(21)+1)/2
```

|style="text-align:center;"|***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;]
||?
|| 2.79128784747792000329402359686400424 
|-

|0.70710 67811 86547 52440  +0.70710 67811 86547 524 i
||Square root of *i* 
||
|bgcolor=#e0f0f0 align=center| \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) 
||
```mathematica
(1+i)/(sqrt 2)
```

|style="text-align:center;"|***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,,...]  [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,,...] *i*
||?
|| 0.70710678118654752440084436210484903    + 0.70710678118654752440084436210484 *i* 
|-

|0.80939 40205 40639 13071
||Alladi–Grinstead constant
||
|bgcolor=#e0f0f0 align=center|{\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)} 
|| 
```mathematica
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
||Sierpiński's constant 
||
|bgcolor=#e0f0f0 align=center| {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)
||
```mathematica
-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
||Reciprocal of the Euler–Mascheroni constant
||
|bgcolor=#e0f0f0 align=center|\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 
||
```mathematica
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
||Lebesgue constant (interpolation)
||
|bgcolor=#e0f0f0 align=center|{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} 
||
```mathematica
1/3 + 2*sqrt(3)/pi
```

|style="text-align:center;"|***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
||Silver root  Tutte–Beraha constant
||
|bgcolor=#e0f0f0 align=center| \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}}}}
||
```mathematica
2+2 cos(2Pi/7)
```

|style="text-align:center;"|***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
||Euler Totient  constant
||
|bgcolor=#e0f0f0 align=center|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} 
||
```mathematica
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
||
|bgcolor=#e0f0f0 align=center|\sqrt[4]{5} 
|| \sqrt[5]{5 \,\sqrt[5]{5 \, \sqrt[5]{5 \,\sqrt[5]{5 \,\sqrt[5]{5 \,\cdots}}}}} 
||
```mathematica
(5(5(5(5(5(5(5) 
 ^1/5)^1/5)^1/5) 
 ^1/5)^1/5)^1/5) 
 ^1/5 ...
```

|style="text-align:center;"|***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
||Area of Ford circle 
||
|bgcolor=#e0f0f0 align=center| A_{CF} 
|| 
\sum_{q\ge 1} \sum_{ (p, q)=1 \atop 1 \le p 
||
```mathematica
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
||Zeta(4)
||
|bgcolor=#e0f0f0 align=center|\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} + ... 
||
```mathematica
Sum[n=1 to ∞]
{1/n^4}
```

|style="text-align:center;"|***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. 
||
|bgcolor=#e0f0f0 align=center|{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 
||
```mathematica
(sqrt(17)-1)/2
```

|style="text-align:center;"|***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;]
||
|| 1.56155281280883027491070492798703851 
|-

|9.86960 44010 89358 61883
||Pi Squared
||
|bgcolor=#e0f0f0 align=center|{\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
||
```mathematica
6 Sum[n=1 to ∞]
{1/n^2}
```

|style="text-align:center;"|***T***
||A002388
||[9;1,6,1,2,47,1,8,1,1,2,2,1,1,8,3,1,10,5,1,3,...]
||
|| 9.86960440108935861883449099987615114 
|-

|1.32471 79572 44746 02596
||Plastic number
||
|bgcolor=#e0f0f0 align=center|{\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}}}
||
```mathematica
(1+(1+(1+(1+(1+(1)
^(1/3))^(1/3))^(1/3))
^(1/3))^(1/3))^(1/3)
```

|style="text-align:center;"|***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
||
|bgcolor=#e0f0f0 align=center|2\,ln\,\gamma 
||\frac{\pi^2}{6ln(2)} 
||
```mathematica
Pi^(2)/(6*ln(2))
```

|style="text-align:center;"|***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
||Regular paperfolding sequence
||
|bgcolor=#e0f0f0 align=center|{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}}}} 
|| 
```mathematica
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
||Cubic recurrence constant
||
|bgcolor=#e0f0f0 align=center|{\sigma_3}
||\prod_{n=1}^\infty n^{{3}^{-n}} = \sqrt[3] {1 \sqrt[3] {2 \sqrt[3]{3 \cdots}}} = 1^{1/3} \; 2^{1/9} \; 3^{1/27} \cdots 
||
```mathematica
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
||Fractal dimension of the Koch snowflake 
||
|bgcolor=#e0f0f0 align=center|{C_k}
|| \frac{\log 4}{\log 3} 
||
```mathematica
log(4)/log(3)
```

|style="text-align:center;"|***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
||
|bgcolor=#e0f0f0 align=center|2^{\,e} 
||2^e 
||
```mathematica
2^e
```

||
||
||[6;1,1,2,1,1,2,3,1,14,11,4,3,1,1,7,5,5,2,7,...]
||
|| 6.58088599101792097085154240388648649 
|-

|0.26149 72128 47642 78375
||Meissel-Mertens constant
||
|bgcolor=#e0f0f0 align=center|{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)}
||
```mathematica
gamma+ 
 Sum[n=1 to ∞] 
 {ln(1-1/prime(n)) 
 +1/prime(n)}
```

|style="text-align:center;"|***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
||
| bgcolor=#e0f0f0 align=center| \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!}}
||
```mathematica
e^(π/2)
```

|style="text-align:center;"|***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
||
|bgcolor=#e0f0f0 align=center|\sqrt[3]{1}
|| \begin{cases} \ \ 1 \\ -\frac{1}{2}+\frac{\sqrt{3}}{2}i \\ -\frac{1}{2}-\frac{\sqrt{3}}{2}i. \end{cases} 
||
```mathematica
1, 
 E^(2i pi/3), 
 E^(-2i pi/3)
```

|style="text-align:center;"|***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, ] i
||
|| - 0.5  ± 0.8660254037844386467637231707529 ***i***  
|-

| 0.11000 10000 00000 00000 0001 
||Liouville number
||
|bgcolor=#e0f0f0 align=center|\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
||
```mathematica
Sum[n=1 to ∞] 
 {10^(-n!)}
```

|style="text-align:center;"|***T***
||
||[1;9,1,999,10,9999999999999,1,9,999,1,9]
||
|| 0.11000100000000000000000100... 
|-

|0.06598 80358 45312 53707
||Lower limit of Tetration
||
|bgcolor=#e0f0f0 align=center|{e}^{-e}
||\left(\frac {1}{e}\right)^e
||
```mathematica
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
||
|bgcolor=#e0f0f0 align=center| {\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}
||
```mathematica
(1/3)*(1+(19+3 
 *sqrt(33))^(1/3) 
 +(19-3 
 *sqrt(33))^(1/3))
```

|style="text-align:center;"|***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
||
|bgcolor=#e0f0f0 align=center|{ll_2}
||-\ln(\ln(2)) 
||
```mathematica
-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
||
|bgcolor=#e0f0f0 align=center|\pi ^\pi
||\pi ^\pi
||
```mathematica
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
||
|bgcolor=#e0f0f0 align=center|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!} 
||
```mathematica
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
||Exponential reiterated constant
||
|bgcolor=#e0f0f0 align=center|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}} 
||
```mathematica
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
||Masser–Gramain constant
||
|bgcolor=#e0f0f0 align=center|{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}
||
```mathematica
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
||The ratio of a square and circle circumscribed
||
|bgcolor=#e0f0f0 align=center|\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}
||
```mathematica
sum[n=1 to ∞]
{(-1)^(floor(
(n-1)/2))
/(2n-1)}
```

|style="text-align:center;"|***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
||Backhouse's constant
||
|bgcolor=#e0f0f0 align=center|{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
||
```mathematica
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
||Gibbs constant
||
|bgcolor=#e0f0f0 align=center|{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 
||
```mathematica
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
||Copeland–Erdős constant
||
|bgcolor=#e0f0f0 align=center|{\mathcal{C}_{CE}}
||\sum _{n=1}^\infty \frac{p_n} {10^{n + \sum \limits_{k=1}^n \lfloor \log_{10}{p_k} \rfloor }} 
||
```mathematica
sum[n=1 to ∞] 
 {prime(n) /(n+(10^ 
 sum[k=1 to n]{floor 
 (log_10 prime(k))}))}
```

|style="text-align:center;"|***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
||Fractal dimension of the boundary of the dragon curve
||
|bgcolor=#e0f0f0 align=center|{C_d}
||\frac{\log\left(\frac{1+\sqrt[3]{73-6\sqrt{87}}+\sqrt[3]{73+6\sqrt{87}}}{3}\right)}
{\log(2)}
||
```mathematica
(log((1+(73-6 sqrt(87))^1/3+ 
(73+6 sqrt(87))^1/3)/3))/ 
log(2)))
```

|style="text-align:center;"|***T***
||
||[1;1,1,10,12,2,1,149,1,1,1,3,11,1,3,17,4,1,...]
||
|| 1.52362708620249210627768393595421662  
|-

|1.78221 39781 91369 11177
||Grothendieck constant
||
|bgcolor=#e0f0f0 align=center|{K_{R}}
|| \frac {\pi}{2 \log(1+\sqrt{2})} 
||
```mathematica
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
||Hausdorff dimension, Sierpinski triangle
||
|bgcolor=#e0f0f0 align=center|{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} 
||
```mathematica
(Sum[n=0 to ∞] {1/
(2^(2n+1) (2n+1))})/ 
 (Sum[n=0 to ∞] {1/
(3^(2n+1) (2n+1))})
```

|style="text-align:center;"|***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
||Mills' constant
||
|bgcolor=#e0f0f0 align=center|{\theta}
|| \lfloor \theta^{3^{n}} \rfloor primes
||
```mathematica
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
||Figure eight knot hyperbolic volume
||
|bgcolor=#e0f0f0 align=center|{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\}

||
```mathematica
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
||Hermite–Ramanujan constant
||
|bgcolor=#e0f0f0 align=center|{R} 
|| e^{\pi\sqrt{163}}
||
```mathematica
e^(π sqrt(163))
```

|style="text-align:center;"|***T***
||
||[262537412640768743;1,1333462407511,1,8,1,1,5,...]
||1859
|| 262537412640768743.999999999999250073  
|-

|1.74540 56624 07346 86349
||Khinchin harmonic mean
||
|bgcolor=#e0f0f0 align=center|{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}}
*a*1 ... *a**n* are elements of a continued fraction [*a*0; *a*1, *a*2, ..., *a**n*]
||
```mathematica
(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
||Square root of the number e
||
|bgcolor=#e0f0f0 align=center|\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
||
```mathematica
Sum[n=0 to ∞]
{1/(2^n n!)}
```

|style="text-align:center;"|***T***
||
||[1;1,1,1,5,1,1,9,1,1,13,1,1,17,1,1,21,1,1,...]  = [1;1,], p∈ℕ
||
|| 1.64872127070012814684865078781416357 
|-

|1.01734 30619 84449 13971
||Zeta(6)
||
|bgcolor=#e0f0f0 align=center|\zeta(6)
||\frac{\pi^6}{945} \! = \! \prod_{n=1}^\infty \! \underset{p_n: \text{ prime}}{ \frac{1}{{1-p_n}^{-6}}} = \frac{1}{1 \! -\! 2^{-6}} \! \cdot \! \frac{1}{1 \! - \! 3^{-6}} \! \cdot \! \frac{1}{1 \! - \! 5^{-6}} \cdots
||
```mathematica
Prod[n=1 to ∞]
{1/(1-ithprime
(n)^-6)}
```
 
|style="text-align:center;"|***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
||Trott constant
||
|bgcolor=#e0f0f0 align=center|\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
||Chaitin constant
||
|bgcolor=#e0f0f0 align=center|\Omega 
||\sum_{p \in P} 2^{-|p|}
* : Halted program
* : Size in bits of program 
* : Domain of all programs that stop.

||
|style="text-align:center;"|***T***
||
||[0; 126, 1, 62, 5, 5, 3, 3, 21, 1, 4, 1]
||1975
||0.0078749969978123844 
|-

|0.83462 68416 74073 18628
||Gauss constant
||
|bgcolor=#e0f0f0 align=center|{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}}
AGM = Arithmetic–geometric mean
||
```mathematica
(4 sqrt(2)((1/4)!)^2)
/pi^(3/2)
```

|style="text-align:center;"|***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
||Ramanujan–Soldner constant
||
|bgcolor=#e0f0f0 align=center|{\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
||
```mathematica
FindRoot[li(x) = 0]
```

|style="text-align:center;"|***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
||Cahen's constant
||
|bgcolor=#e0f0f0 align=center|\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 *k*th 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 
||
|style="text-align:center;"|***T***
||
||[0; 1, 1, 1, 4, 9, 196, 16641, 639988804, ...]
||1891
| 0.64341054628833802618225430775756476  
|-

|1.41421 35623 73095 04880
||Square root of 2, Pythagoras constant.
||
|bgcolor=#e0f0f0 align=center|\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 
||
```mathematica
prod[n=1 to ∞] 
 {1+(-1)^(n+1) 
 /(2n-1)}
```

|style="text-align:center;"|***A***
||
||[1;2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,...] = [1;...]
||
|| 1.41421356237309504880168872420969808  
|-

|1.77245 38509 05516 02729
||Carlson–Levin constant
||Musical interval between each half tone
||

|bgcolor=#e0f0f0 align=center|\sqrt[12]{2}
||
\begin{array}{l|ccccccccccccr}
\! 2^\frac{x}{12} \! & \!\!\scriptstyle{0} & \!\!\!\!\scriptstyle{1} & \!\!\scriptstyle{2} & \!\!\scriptstyle{3}
 & \!\!\scriptstyle{4} & \scriptstyle{5} & \!\!\scriptstyle{6} & \!\!\scriptstyle{7} & \!\!\scriptstyle{8}
 & \!\!\scriptstyle{9} & \!\! \scriptstyle{10} & \!\! \scriptstyle{11} & \!\! \scriptstyle{12} \\
\hline
\! \scriptstyle{\textrm{Key}} \! & \!\scriptstyle{\mathrm{C_1}} & \!\!\scriptstyle{\mathrm{C^\#}} & \!\!\scriptstyle{\mathrm{D}} 
 & \!\scriptstyle{\mathrm{D^\#}} & \!\!\scriptstyle{\mathrm{E}} & \scriptstyle{\mathrm{F}} & \!\scriptstyle{\mathrm{F^\#}}
 & \!\!\scriptstyle{\mathrm{G}} & \!\scriptstyle{\mathrm{G^\#}} & \!\!\scriptstyle{\mathrm{A}} & \!\scriptstyle{\mathrm{A^\#}} 
 & \!\!\scriptstyle{\mathrm{B}} & \!\scriptstyle{\mathrm{C_2}}
\end{array}
(A = 440&nbsp;Hz)
||
```mathematica
2^(1/12)
```

|style="text-align:center;"|***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
||Gieseking constant
||
|bgcolor=#e0f0f0 align=center|{\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).
||
```mathematica
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
||Lemniscate constant
||
|bgcolor=#e0f0f0 align=center|{\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
|| 
```mathematica
4 sqrt(2/pi)
((1/4)!)^2
```

|style="text-align:center;"|***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
||Glaisher–Kinkelin constant 
||
|bgcolor=#e0f0f0 align=center|{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)}
||
```mathematica
e^(1/12-zeta´{-1})
```

|style="text-align:center;"|***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
||Digamma (1/4)
||
|bgcolor=#e0f0f0 align=center|{\psi} (\tfrac14) 
||  -\gamma -\frac{\pi}{2} - 3\ln{2} = -\gamma+\sum_{n=0}^{\infty}\left(\frac{1}{n+1}-\frac{1}{n+\tfrac14}\right)
||
```mathematica
-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
||Strongly Carefree constant
||
|bgcolor=#e0f0f0 align=center|K_{2}
|| \prod_{n=1}^\infty \underset{p_n: \text{ prime}} {\left(1-\frac{3 p_n-2}{{p_n}^{3}}\right)} = \frac {6}{\pi ^2}\prod_{n=1}^\infty \underset{p_n: \text{ prime}} {\left(1-\frac{1}\right)}  
||
```mathematica
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  
|-

|3.62560 99082 21908 31193
||Gamma(1/4)
||
|bgcolor=#e0f0f0 align=center|\Gamma(\tfrac14)
|| 4 \left(\frac{1}{4}\right)! = \left(-\frac{3}{4}\right)! 
||
```mathematica
4(1/4)!
```

|style="text-align:center;"|***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
||Somos' quadratic recurrence constant
||
|bgcolor=#e0f0f0 align=center|{\sigma}
||\prod_{n=1}^\infty n^{{1/2}^n} = \sqrt {1 \sqrt {2 \sqrt{3 \cdots}}} = 1^{1/2} \; 2^{1/4} \; 3^{1/8} \cdots  
||
```mathematica
prod[n=1 to ∞]
{n ^(1/2)^n}
```

|style="text-align:center;"|***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
||
|bgcolor=#e0f0f0 align=center| {\theta_m} 
|| \arctan \left(\sqrt{2}\right) = \arccos \left(\sqrt{\tfrac13}\right) \approx \textstyle {54.7356} ^{ \circ } 
||
```mathematica
arctan(sqrt(2))
```

|style="text-align:center;"|***T***
||
||[0;1,21,2,1,1,1,2,1,2,2,4,1,2,9,1,2,1,1,1,3,...]
||
|| 0.95531661812450927816385710251575775  
|-

|1.78107 24179 90197 98523
||Exp.gamma,  Barnes G-function
||
|bgcolor=#e0f0f0 align=center|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 
||
```mathematica
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  
|-

|0.74759 79202 53411 43517
||Rényi's Parking Constant
|| 
|bgcolor=#e0f0f0 align=center|{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}  
||
```mathematica
[e^(-2*Gamma)] 
*।nt{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  
|-

|1.27323 95447 35162 68615
||Ramanujan–Forsyth series
||
|bgcolor=#e0f0f0 align=center|\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}
||
```mathematica
Sum[n=0 to ∞] 
 {[(2n-3)!! 
 /(2n)!!]^2}
```

|style="text-align:center;"|***I***
||
||[1;3,1,1,1,15,2,72,1,9,1,17,1,2,1,5,1,1,10,...]
||
|| 1.27323954473516268615107010698011489  
|-

|1.44466 78610 09766 13365
||Steiner number, Iterated exponential Constant
||
|bgcolor=#e0f0f0 align=center|\sqrt[e]{e}
||e^{\frac{1}{e}}{\color{White}{...........}} = Upper Limit of Tetration
||
```mathematica
e^(1/e)
```

|style="text-align:center;"|***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
||Minimum value of función  ƒ(x) = xx
||
|bgcolor=#e0f0f0 align=center| {\left(\frac{1}{e}\right)}^\frac{1}{e}
||{e}^{-\frac{1}{e}} {\color{White}{..........}} =।nverse Steiner Number 
||
```mathematica
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
||Pólya Random walk constant
||
|bgcolor=#e0f0f0 align=center|{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}
||
```mathematica
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
||Bloch–Landau constant
||
|bgcolor=#e0f0f0 align=center|{L}
|| = \frac {\Gamma(\tfrac13)\;\Gamma(\tfrac{5}{6})} {\Gamma(\tfrac{1}{6})} = \frac {(-\tfrac23)!\;(-1+\tfrac56)!} {(-1+\tfrac16)!}
||
```mathematica
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
||MRB Constant, Marvin Ray Burns
||
|bgcolor=#e0f0f0 align=center| C_{{}_{MRB}}
|| \sum_{n=1}^{\infty} (-1)^n (n^{1/n}-1) = - \sqrt[1]{1} + \sqrt[2]{2} - \sqrt[3]{3} + \cdots
||
```mathematica
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  
|-

|1.46707 80794 33975 47289
||Porter Constant
||
|bgcolor=#e0f0f0 align=center|{C}
|| \frac{6\ln 2}{\pi ^2} \left(3 \ln 2 + 4 \,\gamma -\frac{24}{\pi ^2} \,\zeta '(2)-2 \right)-\frac{1}{2} 
 \scriptstyle \gamma \, \text{= Euler–Mascheroni Constant} = 0.5772156649\ldots 
 \scriptstyle \zeta '(2) \,\text{= Derivative of }\zeta(2)= 
- \sum \limits_{n = 2}^{\infty} \frac{\ln n}{n^2} = -0.9375482543\ldots 
|| 
```mathematica
6*ln2/pi^2(3*ln2+ 
4 EulerGamma- 
WeierstrassZeta'(2) 
*24/pi^2-2)-1/2
```
 
||
||
||[1;2,7,10,1,2,38,5,4,1,4,12,5,1,5,1,2,3,1,...]
||1974
|| 1.46707807943397547289779848470722995  
|-

|4.66920 16091 02990 67185
||Feigenbaum constant δ
||
|bgcolor=#e0f0f0 align=center|{\delta}
|| \lim_{n \to \infty}\frac {x_{n+1}-x_n}{x_{n+2}-x_{n+1}} \qquad \scriptstyle x \in (3.8284;\, 3.8495)
 \scriptstyle x_{n+1}=\,ax_n(1-x_n)\quad \text{or} \quad x_{n+1}=\,a\sin(x_n)
||
|style="text-align:center;"|***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
||Feigenbaum constant α
||
|bgcolor=#e0f0f0 align=center|\alpha
||\lim_{n \to \infty}\frac {d_n}{d_{n+1}}
||
|style="text-align:center;"|***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
||Golomb–Dickman constant
||
|bgcolor=#e0f0f0 align=center|{\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}
||
```mathematica
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
||Gelfond constant
||
|bgcolor=#e0f0f0 align=center|{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
||
```mathematica
Sum[n=0 to ∞] 
 {(pi^n)/n!}
```

|style="text-align:center;"|***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
||
|bgcolor=#e0f0f0 align=center|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
|| 
```mathematica
Sum[n=0 to ∞]
{2^n/n!}
```

|style="text-align:center;"|***T***
||
||[7;2,1,1,3,18,5,1,1,6,30,8,1,1,9,42,11,1,...]= [7,2,], n = 3, 6, 9, etc.
||
|| 7.38905609893065022723042746057500781  
|-

|0.35323 63718 54995 98454
||Hafner–Sarnak–McCurley constant (1)
||
|bgcolor=#e0f0f0 align=center|{\sigma}
|| \prod_{k=1}^{\infty}\left\{1-[1-\prod_{j=1}^n \underset{p_k: \text{ prime}}{(1-p_k^{-j})]^2}\right\}
||
```mathematica
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
||Hafner–Sarnak–McCurley constant (2)
||
|bgcolor=#e0f0f0 align=center|\frac{1}{\zeta(2)}
|| \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
||
```mathematica
Prod{n=1 to ∞}
(1-1/ithprime(n)^2)
```
 
|style="text-align:center;"|***T***
||
||[0;1,1,1,1,4,2,4,7,1,4,2,3,4,10,1,2,1,1,1,...] 
||
|| 0.60792710185402662866327677925836583  
|-

|0.12345 67891 01112 13141
||Champernowne constant
||
|bgcolor=#e0f0f0 align=center|C_{10}
||\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)}}
||
|style="text-align:center;"|***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
||Landau-Ramanujan constant
||
|bgcolor=#e0f0f0 align=center|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}}
||
|style="text-align:center;"|***T*** ?
||
||[0;1,3,4,6,1,15,1,2,2,3,1,23,3,1,1,3,1,1,6,4,...]
||
|| 0.76422365358922066299069873125009232  
|-

|2.71828 18284 59045 23536
||Number e, Euler's number
||
|bgcolor=#e0f0f0 align=center|{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 
||
```mathematica
Sum[n=0 to ∞]
{1/n!} 
(* lim_(n->∞) 
(1+1/n)^n *)
```

|style="text-align:center;"|***T***
||
||[2;1,2,1,1,4,1,1,6,1,1,8,1,1,10,1,1,12,1,...]  = [2;], p∈ℕ
||
|| 2.71828182845904523536028747135266250  
|-

|0.36787 94411 71442 32159
||Inverse of Number e
||
|bgcolor=#e0f0f0 align=center|\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

||
```mathematica
Sum[n=2 to ∞]
{(-1)^n/n!}
```

|style="text-align:center;"|***T***
||
||[0;2,1,1,2,1,1,4,1,1,6,1,1,8,1,1,10,1,1,12,...]  = [0;2,1,], p∈ℕ
||1618
|| 0.36787944117144232159552377016146086  
|-

|0.69034 71261 14964 31946
||Upper iterated exponential
||
|bgcolor=#e0f0f0 align=center| {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^}}}}  
||
```mathematica
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
||
 |bgcolor=#e0f0f0 align=center| {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^}}}}  
||
```mathematica
2^-3^-4^-5^-6^ 
 -7^-8^-9^-10^ 
 -11^-12 …
```

||
||
||[0;1,1,1,12,1,2,1,1,4,3,1,1,2,1,2,1,51,2,2,1,...]
||
||0.6583655992... 
|-

|3.14159 26535 89793 23846
||π number, Archimedes number
||
|bgcolor=#e0f0f0 align=center| \pi 
||\lim_{n\to \infty }\, 2^n \underbrace{\sqrt{2-\sqrt{2+\sqrt{2+\cdots +\sqrt{2}}}}}_n
||
```mathematica
Sum[n=0 to ∞]
{(-1)^n 4/(2n+1)}
```
 
|style="text-align:center;"|***T***
||
||[3;7,15,1,292,1,1,1,2,1,3,1,14,2,1,1,2,2,2,...]
||
|| 3.14159265358979323846264338327950288  
|-

|1.92878 00...
||Wright constant
||
|bgcolor=#e0f0f0 align=center|{\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...
|-

|0.46364 76090 00806 11621
||Machin–Gregory series
||
|bgcolor=#e0f0f0 align=center|\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}
||
```mathematica
Sum[n=0 to ∞] 
{(-1)^n (1/2)^(2n+1)
/(2n+1)}
```

|style="text-align:center;"|***I***
||
||[0;2,6,2,1,1,1,6,1,2,1,1,2,10,1,2,1,2,1,1,1,...]
||
|| 0.46364760900080611621425623146121440  
|-

|0.69777 46579 64007 98200
||Continued fraction constant, Bessel function
||
|bgcolor=#e0f0f0 align=center|{C}_{CF}
|| \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}}}}}}} 

||
```mathematica
(Sum [n=0 to ∞]
{n/(n!n!)}) /
(Sum [n=0 to ∞]
{1/(n!n!)})
```

|style="text-align:center;"|***I***
||
||[0;1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,...]  = [0;], p∈ℕ

||
|| 0.69777465796400798200679059255175260  
|-

|1.90216 05831 04
||Brun 2  constant = Σ inverse of Twin primes
||
|bgcolor=#e0f0f0 align=center|{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
||Brun 4  constant = Σ inv.prime quadruplets
||
|bgcolor=#e0f0f0 align=center|{B}_{\,4}
||\textstyle {\sum(\frac1{p}+\frac1{p+2}+\frac1{p+6}+\frac1{p+8})} \scriptstyle \quad {p,\; p+2,\; p+6,\; p+8: \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

||Buffon constant
||
|bgcolor=#e0f0f0 align=center|\frac{2}{\pi}
|| \frac{\sqrt2}2 \cdot \frac{\sqrt{2+\sqrt2}}2 \cdot \frac{\sqrt{2+\sqrt{2+\sqrt2}}}2 \cdots Viète product
||
```mathematica
2/Pi
```

|style="text-align:center;"|***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
||Euler–Gompertz constant
||
|bgcolor=#e0f0f0 align=center|{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}} }}}}}} 

||
```mathematica
integral[0 to ∞]
{(e^-n)/(1+n)}
```

|style="text-align:center;"|***I***
||
||[0;1,1,2,10,1,1,4,2,2,13,2,4,1,32,4,8,1,1,1,...]

||
|| 0.59634736232319407434107849936927937  
|-

| ***i*** ··· 
||Imaginary number
||
|bgcolor=#e0f0f0 align=center|{i}
||\sqrt{-1} = \frac{\ln(-1)}{\pi} \qquad\qquad \mathrm{e}^{i\,\pi} = -1
||
```mathematica
sqrt(-1)
```

|style="text-align:center;"|***C I***
|| 
||
||1501  to  1576
|| ***i*** 
|-

|2.74723 82749 32304 33305
||Ramanujan nested radical
||
|bgcolor=#e0f0f0 align=center| 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}
||
```mathematica
(2+sqrt(5)
+sqrt(15
-6 sqrt(5)))/2
```

|style="text-align:center;"|***A***
||
||[2;1,2,1,21,1,7,2,1,1,2,1,2,1,17,4,4,1,1,4,2,...]
||
|| 2.74723827493230433305746518613420282  
|-

|0.56714 32904 09783 87299
||Omega constant, Lambert W function
||
|bgcolor=#e0f0f0 align=center|{\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^}}}}  
||
```mathematica
Sum[n=1 to ∞]
{(-n)^(n-1)/n!}
```

|style="text-align:center;"|***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)
||
|bgcolor=#e0f0f0 align=center|{\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 
||
```mathematica
Sum[n=1 to ∞]
{(-1)^(n+1)
/(-1+2n)^3}
```

|style="text-align:center;"|***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
||
|bgcolor=#e0f0f0 align=center| \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}
||
```mathematica
Sum[k=0 to 4]
{e^(2k^2 pi i/5)}
```

|style="text-align:center;"|***A***
||
||[2;4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,...]  = [2;,...]
||
|| 2.23606797749978969640917366873127624  
|-

|3.35988 56662 43177 55317
||Prévost constant Reciprocal Fibonacci constant
||
|bgcolor=#e0f0f0 align=center| \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
*Fn*: Fibonacci series
||
```mathematica
Sum[n=1 to ∞]
{1/Fibonacci[n]}
```

|style="text-align:center;"|***I***
||
||[3;2,1,3,1,1,13,2,3,3,2,1,1,6,3,2,4,362,...]
||?
|| 3.35988566624317755317201130291892717  
|-

|
||Khinchin's constant
||
|bgcolor=#e0f0f0 align=center| K_{\,0} 
|| \prod_{n=1}^\infty \left[{1+{1\over n(n+2)}}\right]^{\ln n/\ln 2}
||
```mathematica
Prod[n=1 to ∞] 
 {(1+1/(n(n+2))) 
 ^(ln(n)/ln(2))}
```

|style="text-align:center;"|***T***
||
||[2;1,2,5,1,1,2,1,1,3,10,2,1,3,2,24,1,3,2,...]
||1934
|| 2.68545200106530644530971483548179569  
|-

|}

## ਇਹ ਵੀ ਦੇਖੋ 
*ਇਨਵੇਰੀਅੰਟ (ਗਣਿਤ)
*ਸੰਖਿਆਵਾਂ ਦੀ ਸੂਚੀ
*ਗਣਿਤਿਕ ਸਥਿਰਾਂਕ

## ਬਾਹਰੀ ਲਿੰਕ 
* Symbolic Calculator, Plouffe's।nverter
* Constants - from Wolfram MathWorld
* On-Line Encyclopedia of।nteger Sequences (OEIS)
* Steven Finch's page of mathematical constants
* Xavier Gourdon and Pascal Sebah's page of numbers, mathematical constants and algorithms

## ਨੋਟਸ 
### Site MathWorld Wolfram.com 
### Site OEIS Wiki 
*
ਸਥਿਰਾਂਕ
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