Mathematical constant

A mathematical constant is a number, which has a special meaning for calculations. For example, the constant π (pronounced "pie") means the ratio of a circle's circumference to its diameter.[1][2] This value is always the same for any circle. A mathematical constant is often a real, non-integral number of interest.[3]

In contrast to physical constants, mathematical constants do not come from physical measurements.

Key mathematical constants

The following table contains some important mathematical constants:

Name Symbol Value Meaning
Pi, Archimedes' constant or Ludoph's number π ≈3.141592653589793 A transcendental number that is the ratio of the length of a circle's circumference to its diameter. It is also the area of the unit circle.
e, Napier's constant or Euler's (pronounced "oilers") number e ≈2.718281828459045 A transcendental number that is the base of natural logarithms.
Phi, golden ratio φ ≈1.618033988749894 It is the value of a larger value divided by a smaller value if this is equal to the value of the sum of the values divided by the larger value.
Square root of 2, Pythagoras' constant √2 ≈1.414213562373095 An irrational number that is the length of the diagonal of a square with sides of length 1. This number can not be written as a fraction.

Constants and series

The following table contains a list of constants and series in mathematics, with the following columns:

• Value: Numerical value of the constant.
• LaTeX: Formula or series in TeX format.
• Formula: For use in programs such as Mathematica or Wolfram Alpha.
• OEIS: Link to On-Line Encyclopedia of Integer Sequences (OEIS), where the constants are available with more details.
• Continued fraction: In the simple form [to integer; frac1, frac2, frac3, ...] (in brackets if periodic)
• Type:

Note that the list can be ordered correspondingly by clicking on the header title at the top of the table.

Value Name Symbol LaTeX Formula Type OEIS Continued fraction
3.24697960371746706105000976800847962 Silver, Tutte–Beraha constant ${\displaystyle \varsigma }$ ${\displaystyle 2+2\cos(2\pi /7)=\textstyle 2+{\frac {2+{\sqrt[{3}]{7+7{\sqrt[{3}]{7+7{\sqrt[{3}]{\,7+\cdots }}}}}}}{1+{\sqrt[{3}]{7+7{\sqrt[{3}]{7+7{\sqrt[{3}]{\,7+\cdots }}}}}}}}}$ 2+2 cos(2Pi/7) T A116425 [3;4,20,2,3,1,6,10,5,2,2,1,2,2,1,18,1,1,3,2,...]
1.09864196439415648573466891734359621 Paris constant ${\displaystyle C_{Pa}}$ ${\displaystyle \prod _{n=2}^{\infty }{\frac {2\varphi }{\varphi +\varphi _{n}}}\;,\varphi ={Fi}}$ I A105415 [1;10,7,3,1,3,1,5,1,4,2,7,1,2,3,22,1,2,5,2,1,...]
2.74723827493230433305746518613420282 Ramanujan nested radical R5 ${\displaystyle R_{5}}$ ${\displaystyle \scriptstyle {\sqrt {5+{\sqrt {5+{\sqrt {5-{\sqrt {5+{\sqrt {5+{\sqrt {5+{\sqrt {5-\cdots }}}}}}}}}}}}}}\;=\textstyle {\frac {2+{\sqrt {5}}+{\sqrt {15-6{\sqrt {5}}}}}{2}}}$ (2+sqrt(5)+sqrt(15-6 sqrt(5)))/2 I [2;1,2,1,21,1,7,2,1,1,2,1,2,1,17,4,4,1,1,4,2,...]
2.23606797749978969640917366873127624 Square root of 5, Gauss sum ${\displaystyle {\sqrt {5}}}$ ${\displaystyle \scriptstyle \forall \,n=5,\displaystyle \sum _{k=0}^{n-1}e^{\frac {2k^{2}\pi i}{n}}=1+e^{\frac {2\pi i}{5}}+e^{\frac {8\pi i}{5}}+e^{\frac {18\pi i}{5}}+e^{\frac {32\pi i}{5}}}$ Sum[k=0 to 4]{e^(2k^2 pi i/5)} I A002163 [2;4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,...]
= [2;(4),...]
3.62560990822190831193068515586767200 Gamma(1/4) ${\displaystyle \Gamma ({\tfrac {1}{4}})}$ ${\displaystyle 4\left({\frac {1}{4}}\right)!=\left(-{\frac {3}{4}}\right)!}$ 4(1/4)! T A068466 [3;1,1,1,2,25,4,9,1,1,8,4,1,6,1,1,19,1,1,4,1,...]
0.18785964246206712024851793405427323 MRB constant, Marvin Ray Burns ${\displaystyle C_{_{MRB}}}$ ${\displaystyle \sum _{n=1}^{\infty }({-}1)^{n}(n^{1/n}{-}1)=-{\sqrt[{1}]{1}}+{\sqrt[{2}]{2}}-{\sqrt[{3}]{3}}+{\sqrt[{4}]{4}}\,\dots }$ Sum[n=1 to ∞]{(-1)^n (n^(1/n)-1)} T A037077 [0;5,3,10,1,1,4,1,1,1,1,9,1,1,12,2,17,2,2,1,...]
0.11494204485329620070104015746959874 Kepler–Bouwkamp constant ${\displaystyle {\rho }}$ ${\displaystyle \prod _{n=3}^{\infty }\cos \left({\frac {\pi }{n}}\right)=\cos \left({\frac {\pi }{3}}\right)\cos \left({\frac {\pi }{4}}\right)\cos \left({\frac {\pi }{5}}\right)\dots }$ prod[n=3 to ∞]{cos(pi/n)} T A085365 [0;8,1,2,2,1,272,2,1,41,6,1,3,1,1,26,4,1,1,...]
1.78107241799019798523650410310717954 Exp(gamma)
G-Barnes function
${\displaystyle e^{\gamma }}$ ${\displaystyle \prod _{n=1}^{\infty }{\frac {e^{\frac {1}{n}}}{1+{\tfrac {1}{n}}}}=\prod _{n=0}^{\infty }\left(\prod _{k=0}^{n}(k+1)^{(-1)^{k+1}{n \choose k}}\right)^{\frac {1}{n+1}}=}$

${\displaystyle \textstyle \left({\frac {2}{1}}\right)^{1/2}\left({\frac {2^{2}}{1\cdot 3}}\right)^{1/3}\left({\frac {2^{3}\cdot 4}{1\cdot 3^{3}}}\right)^{1/4}\left({\frac {2^{4}\cdot 4^{4}}{1\cdot 3^{6}\cdot 5}}\right)^{1/5}\dots }$

Prod[n=1 to ∞]{e^(1/n)}/{1 + 1/n} T A073004 [1;1,3,1,1,3,5,4,1,1,2,2,1,7,9,1,16,1,1,1,2,...]
1.28242712910062263687534256886979172 Glaisher–Kinkelin constant ${\displaystyle {A}}$ ${\displaystyle e^{{\frac {1}{12}}-\zeta ^{\prime }(-1)}=e^{{\frac {1}{8}}-{\frac {1}{2}}\sum \limits _{n=0}^{\infty }{\frac {1}{n+1}}\sum \limits _{k=0}^{n}\left(-1\right)^{k}{\binom {n}{k}}\left(k+1\right)^{2}\ln(k+1)}}$ e^(1/2-zeta´{-1}) T A074962 [1;3,1,1,5,1,1,1,3,12,4,1,271,1,1,2,7,1,35,...]
7.38905609893065022723042746057500781 Schwarzschild conic constant ${\displaystyle e^{2}}$ ${\displaystyle \sum _{n=0}^{\infty }{\frac {2^{n}}{n!}}=1+2+{\frac {2^{2}}{2!}}+{\frac {2^{3}}{3!}}+{\frac {2^{4}}{4!}}+{\frac {2^{5}}{5!}}+\dots }$ Sum[n=0 to ∞]{2^n/n!} T A072334 [7;2,1,1,3,18,5,1,1,6,30,8,1,1,9,42,11,1,...]
= [7,2,(1,1,n,4*n+6,n+2)], n = 3, 6, 9, etc.
1.01494160640965362502120255427452028 Gieseking constant ${\displaystyle {G_{Gi}}}$ ${\displaystyle {\frac {3{\sqrt {3}}}{4}}\left(1-\sum _{n=0}^{\infty }{\frac {1}{(3n+2)^{2}}}+\sum _{n=1}^{\infty }{\frac {1}{(3n+1)^{2}}}\right)=}$

${\displaystyle \textstyle {\frac {3{\sqrt {3}}}{4}}\left(1-{\frac {1}{2^{2}}}+{\frac {1}{4^{2}}}-{\frac {1}{5^{2}}}+{\frac {1}{7^{2}}}-{\frac {1}{8^{2}}}+{\frac {1}{10^{2}}}\pm \dots \right)}$.

T A143298 [1;66,1,12,1,2,1,4,2,1,3,3,1,4,1,56,2,2,11,...]
2.62205755429211981046483958989111941 Lemniscata constant ${\displaystyle {\varpi }}$ ${\displaystyle \pi \,{G}=4{\sqrt {\tfrac {2}{\pi }}}\,({\tfrac {1}{4}}!)^{2}}$ 4 sqrt(2/pi) (1/4!)^2 T A062539 [2;1,1,1,1,1,4,1,2,5,1,1,1,14,9,2,6,2,9,4,1,...]
0.83462684167407318628142973279904680 G, Gauss constant ${\displaystyle {G}}$ ${\displaystyle {\underset {Agm:\;Arithmetic-geometric\;mean}{{\frac {1}{\mathrm {agm} (1,{\sqrt {2}})}}={\frac {4{\sqrt {2}}\,({\tfrac {1}{4}}!)^{2}}{\pi ^{3/2}}}}}}$ (4 sqrt(2)(1/4!)^2)/pi^(3/2) T A014549 [0;1,5,21,3,4,14,1,1,1,1,1,3,1,15,1,3,7,1,...]
1.01734306198444913971451792979092052 Zeta(6) ${\displaystyle \zeta (6)}$ ${\displaystyle {\frac {\pi ^{6}}{945}}=\prod _{n=1}^{\infty }{\underset {p_{n}:\,{primo}}{\frac {1}{{1-p_{n}}^{-6}}}}={\frac {1}{1{-}2^{-6}}}{\cdot }{\frac {1}{1{-}3^{-6}}}{\cdot }{\frac {1}{1{-}5^{-6}}}...}$ Prod[n=1 to ∞] {1/(1-ithprime(n)^-6)} T A013664 [1;57,1,1,1,15,1,6,3,61,1,5,3,1,6,1,3,3,6,1,...]
0,60792710185402662866327677925836583 Constante de Hafner-Sarnak-McCurley ${\displaystyle {\frac {1}{\zeta (2)}}}$ ${\displaystyle {\frac {6}{\pi ^{2}}}{=}\prod _{n=0}^{\infty }{\underset {p_{n}:\,{primo}}{\left(1-{\frac {1}{{p_{n}}^{2}}}\right)}}{=}\textstyle \left(1{-}{\frac {1}{2^{2}}}\right)\left(1{-}{\frac {1}{3^{2}}}\right)\left(1{-}{\frac {1}{5^{2}}}\right)\dots }$ Prod{n=1 to ∞} (1-1/ithprime(n)^2) T A059956 [0;1,1,1,1,4,2,4,7,1,4,2,3,4,10,1,2,1,1,1,...]
1.11072073453959156175397024751517342 The ratio of a square and circumscribed or inscribed circles ${\displaystyle {\frac {\pi }{2{\sqrt {2}}}}}$ ${\displaystyle \sum _{n=1}^{\infty }{\frac {(-1)^{\lfloor {\frac {n-1}{2}}\rfloor }}{2n+1}}={\frac {1}{1}}+{\frac {1}{3}}-{\frac {1}{5}}-{\frac {1}{7}}+{\frac {1}{9}}+{\frac {1}{11}}-\dots }$ sum[n=1 to ∞]{(-1)^(floor((n-1)/2))/(2n-1)} T A093954 [1;9,31,1,1,17,2,3,3,2,3,1,1,2,2,1,4,9,1,3,...]
2.80777024202851936522150118655777293 Fransén–Robinson constant ${\displaystyle {F}}$ ${\displaystyle \int _{0}^{\infty }{\frac {1}{\Gamma (x)}}\,dx.=e+\int _{0}^{\infty }{\frac {e^{-x}}{\pi ^{2}+\ln ^{2}x}}\,dx}$ N[int[0 to ∞] {1/Gamma(x)}] T A058655 [2;1,4,4,1,18,5,1,3,4,1,5,3,6,1,1,1,5,1,1,1...]
1.64872127070012814684865078781416357 Square root of e ${\displaystyle {\sqrt {e}}}$ ${\displaystyle \sum _{n=0}^{\infty }{\frac {1}{2^{n}n!}}=\sum _{n=0}^{\infty }{\frac {1}{(2n)!!}}={\frac {1}{1}}+{\frac {1}{2}}+{\frac {1}{8}}+{\frac {1}{48}}+\cdots }$ sum[n=0 to ∞]{1/(2^n n!)} T A019774 [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
i i, imaginary unit ${\displaystyle {i}}$ ${\displaystyle {\sqrt {-1}}={\frac {\ln(-1)}{\pi }}\qquad \qquad \mathrm {e} ^{i\,\pi }=-1}$ sqrt(-1) C
262537412640768743.999999999999250073 Hermite-Ramanujan constant ${\displaystyle {R}}$ ${\displaystyle e^{\pi {\sqrt {163}}}}$ e^(π sqrt(163)) T A060295 [262537412640768743;1,1333462407511,1,8,1,1,5,...]
4.81047738096535165547303566670383313 John constant ${\displaystyle \gamma }$ ${\displaystyle {\sqrt[{i}]{i}}=i^{-i}=i^{\frac {1}{i}}=(i^{i})^{-1}=e^{\frac {\pi }{2}}}$ e^(π/2) T A042972 [4;1,4,3,1,1,1,1,1,1,1,1,7,1,20,1,3,6,10,3,...]
4.53236014182719380962768294571666681 Constante de Van der Pauw ${\displaystyle \alpha }$ ${\displaystyle {\frac {\pi }{ln(2)}}={\frac {\sum _{n=0}^{\infty }{\frac {4(-1)^{n}}{2n+1}}}{\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n}}}}={\frac {{\frac {4}{1}}{-}{\frac {4}{3}}{+}{\frac {4}{5}}{-}{\frac {4}{7}}{+}{\frac {4}{9}}-\dots }{{\frac {1}{1}}{-}{\frac {1}{2}}{+}{\frac {1}{3}}{-}{\frac {1}{4}}{+}{\frac {1}{5}}-\dots }}}$ π/ln(2) T A163973 [4;1,1,7,4,2,3,3,1,4,1,1,4,7,2,3,3,12,2,1,...]
0.76159415595576488811945828260479359 Hyperbolic tangent (1) ${\displaystyle th\,1}$ ${\displaystyle {\frac {e-{\frac {1}{e}}}{e+{\frac {1}{e}}}}={\frac {e^{2}-1}{e^{2}+1}}}$ (e-1/e)/(e+1/e) T A073744 [0;1,3,5,7,9,11,13,15,17,19,21,23,25,27,...]
= [0;(2p+1)], p∈ℕ
0.69777465796400798200679059255175260 Continued Fraction constant ${\displaystyle {C}_{CF}}$ ${\displaystyle {\underset {J_{k}(){Bessel}}{\underset {Function}{\frac {J_{1}(2)}{J_{0}(2)}}}}={\frac {\sum \limits _{n=0}^{\infty }{\frac {n}{n!n!}}}{\sum \limits _{n=0}^{\infty }{\frac {1}{n!n!}}}}={\frac {{\frac {0}{1}}+{\frac {1}{1}}+{\frac {2}{4}}+{\frac {3}{36}}+{\frac {4}{576}}+\dots }{{\frac {1}{1}}+{\frac {1}{1}}+{\frac {1}{4}}+{\frac {1}{36}}+{\frac {1}{576}}+\dots }}}$ (sum {n=0 to inf} n/(n!n!)) /(sum {n=0 to inf} 1/(n!n!)) A052119 [0;1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,...]
= [0;(p+1)], p∈ℕ
0.36787944117144232159552377016146086 Inverse Napier constant ${\displaystyle {\frac {1}{e}}}$ ${\displaystyle \sum _{n=0}^{\infty }{\frac {(-1)^{n}}{n!}}={\frac {1}{0!}}-{\frac {1}{1!}}+{\frac {1}{2!}}-{\frac {1}{3!}}+{\frac {1}{4!}}-{\frac {1}{5!}}+\dots }$ sum[n=2 to ∞]{(-1)^n/n!} T A068985 [0;2,1,1,2,1,1,4,1,1,6,1,1,8,1,1,10,1,1,12,...]
= [0;2,1,(1,2p,1)], p∈ℕ
2.71828182845904523536028747135266250 Napier constant ${\displaystyle e}$ ${\displaystyle \sum _{n=0}^{\infty }{\frac {1}{n!}}={\frac {1}{0!}}+{\frac {1}{1}}+{\frac {1}{2!}}+{\frac {1}{3!}}+{\frac {1}{4!}}+{\frac {1}{5!}}+\cdots }$ Sum[n=0 to ∞]{1/n!} T A001113 [2;1,2,1,1,4,1,1,6,1,1,8,1,1,10,1,1,12,1,...]
= [2;(1,2p,1)], p∈ℕ
0.49801566811835604271369111746219809
- 0.15494982830181068512495513048388 i
Factorial of i ${\displaystyle i\,!}$ ${\displaystyle \Gamma (1+i)=i\,\Gamma (i)}$ Gamma(1+i) C A212877
A212878
[0;6,2,4,1,8,1,46,2,2,3,5,1,10,7,5,1,7,2,...]
- [0;2,125,2,18,1,2,1,1,19,1,1,1,2,3,34,...] i
0.43828293672703211162697516355126482
+ 0.36059247187138548595294052690600 i
Infinite
Tetration of i
${\displaystyle {}^{\infty }i}$ ${\displaystyle \lim _{n\to \infty }{}^{n}i=\lim _{n\to \infty }\underbrace {i^{i^{\cdot ^{\cdot ^{i}}}}} _{n}}$ i^i^i^... C A077589
A077590
[0;2,3,1,1,4,2,2,1,10,2,1,3,1,8,2,1,2,1, ...]
+ [0;2,1,3,2,2,3,1,5,5,1,2,1,10,10,6,1,1...] i
0.56755516330695782538461314419245334 Module of
Infinite
Tetration of i
${\displaystyle |{}^{\infty }i|}$ ${\displaystyle \lim _{n\to \infty }\left|{}^{n}i\right|=\left|\lim _{n\to \infty }\underbrace {i^{i^{\cdot ^{\cdot ^{i}}}}} _{n}\right|}$ Mod(i^i^i^...) A212479 [0;1,1,3,4,1,58,12,1,51,1,4,12,1,1,2,2,3,...]
0.26149721284764278375542683860869585 Meissel-Mertens constant ${\displaystyle M}$ ${\displaystyle \lim _{n\rightarrow \infty }\left(\sum _{p\leq n}{\frac {1}{p}}-\ln(\ln(n))\right)}$ ..... p: primes A077761 [0;3,1,4,1,2,5,2,1,1,1,1,13,4,2,4,2,1,33,...]
1.9287800... Wright constant ${\displaystyle \omega }$ ${\displaystyle \left\lfloor 2^{2^{2^{\cdot ^{\cdot ^{2^{\omega }}}}}}\right\rfloor }$ = primos: ${\displaystyle \quad }$ ${\displaystyle \left\lfloor 2^{\omega }\right\rfloor }$ =3, ${\displaystyle \left\lfloor 2^{2^{\omega }}\right\rfloor }$ =13, ${\displaystyle \left\lfloor 2^{2^{2^{\omega }}}\right\rfloor }$ =16381, ${\displaystyle \dots }$ A086238 [1; 1, 13, 24, 2, 1, 1, 3, 1, 1, 3]
0.37395581361920228805472805434641641 Artin constant ${\displaystyle C_{Artin}}$ ${\displaystyle \prod _{n=1}^{\infty }\left(1-{\frac {1}{p_{n}(p_{n}-1)}}\right)}$ ...... pn: primo T A005596 [0;2,1,2,14,1,1,2,3,5,1,3,1,5,1,1,2,3,5,46,...]
4.66920160910299067185320382046620161 Feigenbaum constant δ ${\displaystyle {\delta }}$ ${\displaystyle \lim _{n\to \infty }{\frac {x_{n+1}-x_{n}}{x_{n+2}-x_{n+1}}}\qquad \scriptstyle x\in (3,8284;\,3,8495)}$

${\displaystyle \scriptstyle x_{n+1}=\,ax_{n}(1-x_{n})\quad {o}\quad x_{n+1}=\,a\sin(x_{n})}$

T A006890 [4;1,2,43,2,163,2,3,1,1,2,5,1,2,3,80,2,5,...]
2.50290787509589282228390287321821578 Feigenbaum constant α ${\displaystyle \alpha }$ ${\displaystyle \lim _{n\to \infty }{\frac {d_{n}}{d_{n+1}}}}$ T A006891 [2;1,1,85,2,8,1,10,16,3,8,9,2,1,40,1,2,3,...]
5.97798681217834912266905331933922774 Hexagonal Madelung Constant 2 ${\displaystyle H_{2}(2)}$ ${\displaystyle \pi \ln(3){\sqrt {3}}}$ Pi Log[3]Sqrt[3] T A086055 [5;1,44,2,2,1,15,1,1,12,1,65,11,1,3,1,1,...]
0.96894614625936938048363484584691860 Beta(3) ${\displaystyle \beta (3)}$ ${\displaystyle {\frac {\pi ^{3}}{32}}=\sum _{n=1}^{\infty }{\frac {-1^{n+1}}{(-1+2n)^{3}}}={\frac {1}{1^{3}}}{-}{\frac {1}{3^{3}}}{+}{\frac {1}{5^{3}}}{-}{\frac {1}{7^{3}}}{+}\dots }$ Sum[n=1 to ∞]{(-1)^(n+1)/(-1+2n)^3} T A153071 [0;1,31,4,1,18,21,1,1,2,1,2,1,3,6,3,28,1,...]
1.902160583104 Brun constant 2 = Σ inverse twin primes ${\displaystyle B_{\,2}}$ ${\displaystyle \textstyle \sum {\underset {p,\,p+2:\,{primos}}{({\frac {1}{p}}+{\frac {1}{p+2}})}}=({\frac {1}{3}}{+}{\frac {1}{5}})+({\tfrac {1}{5}}{+}{\tfrac {1}{7}})+({\tfrac {1}{11}}{+}{\tfrac {1}{13}})+\dots }$ A065421 [1; 1, 9, 4, 1, 1, 8, 3, 4, 4, 2, 2]
0.870588379975 Brun constant 4 = Σ inverse of twin prime ${\displaystyle B_{\,4}}$ ${\displaystyle {\underset {p,\,p+2,\,p+4,\,p+6:\,{primes}}{\left({\tfrac {1}{5}}+{\tfrac {1}{7}}+{\tfrac {1}{11}}+{\tfrac {1}{13}}\right)}}+\left({\tfrac {1}{11}}+{\tfrac {1}{13}}+{\tfrac {1}{17}}+{\tfrac {1}{19}}\right)+\dots }$ A213007 [0; 1, 6, 1, 2, 1, 2, 956, 3, 1, 1]
22.4591577183610454734271522045437350 pi^e ${\displaystyle \pi ^{e}}$ ${\displaystyle \pi ^{e}}$ pi^e A059850 [22;2,5,1,1,1,1,1,3,2,1,1,3,9,15,25,1,1,5,...]
3.14159265358979323846264338327950288 Pi, Archimedes constant ${\displaystyle \pi }$ ${\displaystyle \lim _{n\to \infty }\,2^{n}\underbrace {\sqrt {2-{\sqrt {2+{\sqrt {2+\dots +{\sqrt {2}}}}}}}} _{n}}$ Sum[n=0 to ∞]{(-1)^n 4/(2n+1)} T A000796 [3;7,15,1,292,1,1,1,2,1,3,1,14,...]
0.06598803584531253707679018759684642 ${\displaystyle e^{-e}}$ ${\displaystyle e^{-e}}$ ... Lower limit of Tetration T A073230 [0;15,6,2,13,1,3,6,2,1,1,5,1,1,1,9,4,1,1,1,...]
0.20787957635076190854695561983497877 i^i ${\displaystyle i^{i}}$ ${\displaystyle e^{\frac {-\pi }{2}}}$ e^(-pi/2) T A049006 [0;4,1,4,3,1,1,1,1,1,1,1,1,7,1,20,1,3,6,10,...]
0.28016949902386913303643649123067200 Bernstein constant ${\displaystyle \beta }$ ${\displaystyle {\frac {1}{2{\sqrt {\pi }}}}}$ T A073001 [0;3,1,1,3,9,6,3,1,3,13,1,16,3,3,4,…]
0.28878809508660242127889972192923078 Flajolet and Richmond ${\displaystyle Q}$ ${\displaystyle \prod _{n=1}^{\infty }\left(1-{\frac {1}{2^{n}}}\right)=\left(1{-}{\frac {1}{2^{1}}}\right)\left(1{-}{\frac {1}{2^{2}}}\right)\left(1{-}{\frac {1}{2^{3}}}\right)\dots }$ prod[n=1 to ∞]{1-1/2^n} A048651
0.31830988618379067153776752674502872 Inverse of Pi, Ramanujan ${\displaystyle {\frac {1}{\pi }}}$ ${\displaystyle {\frac {2{\sqrt {2}}}{9801}}\sum _{n=0}^{\infty }{\frac {(4n)!(1103+26390n)}{(n!)^{4}396^{4n}}}}$ T A049541 [0;3,7,15,292,1,1,1,2,1,3,1,14,2,1,1,...]
0.47494937998792065033250463632798297 Weierstraß constant ${\displaystyle W_{_{WE}}}$ ${\displaystyle {\frac {e^{\frac {\pi }{8}}{\sqrt {\pi }}}{4*2^{3/4}{({\frac {1}{4}}!)^{2}}}}}$ (E^(Pi/8) Sqrt[Pi])/(4 2^(3/4) (1/4)!^2) T A094692 [0;2,9,2,11,1,6,1,4,6,3,19,9,217,1,2,...]
0.56714329040978387299996866221035555 Omega constant ${\displaystyle \Omega }$ ${\displaystyle W(1)=\sum _{n=1}^{\infty }{\frac {(-n)^{n-1}}{n!}}=1{-}1{+}{\frac {3}{2}}{-}{\frac {8}{3}}{+}{\frac {125}{24}}-\dots }$ sum[n=1 to ∞]{(-n)^(n-1)/n!} T A030178 [0;1,1,3,4,2,10,4,1,1,1,1,2,7,306,1,5,1,...]
0.57721566490153286060651209008240243 Euler's number ${\displaystyle \gamma }$ ${\displaystyle -\psi (1)=\sum _{n=1}^{\infty }\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{2^{n}+k}}}$ sum[n=1 to ∞]|sum[k=0 to ∞]{((-1)^k)/(2^n+k)} ? A001620 [0;1,1,2,1,2,1,4,3,13,5,1,1,8,1,2,...]
0.60459978807807261686469275254738524 Dirichlet serie ${\displaystyle {\frac {\pi }{3{\sqrt {3}}}}}$ ${\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n{2n \choose n}}}=1-{\frac {1}{2}}+{\frac {1}{4}}-{\frac {1}{5}}+{\frac {1}{7}}-{\frac {1}{8}}+\cdots }$ Sum[1/(n Binomial[2 n, n]), {n, 1, ∞}] T A073010 [0;1,1,1,1,8,10,2,2,3,3,1,9,2,5,4,1,27,27,...]
0.63661977236758134307553505349005745 2/Pi, François Viète ${\displaystyle {\frac {2}{\pi }}}$ ${\displaystyle {\frac {\sqrt {2}}{2}}\cdot {\frac {\sqrt {2+{\sqrt {2}}}}{2}}\cdot {\frac {\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}{2}}\cdots }$ T A060294 [0;1,1,1,3,31,1,145,1,4,2,8,1,6,1,2,3,1,4,...]
0.66016181584686957392781211001455577 Twin prime constant ${\displaystyle C_{2}}$ ${\displaystyle \prod _{p=3}^{\infty }{\frac {p(p-2)}{(p-1)^{2}}}}$ prod[p=3 to ∞]{p(p-2)/(p-1)^2 A005597 [0;1,1,1,16,2,2,2,2,1,18,2,2,11,1,1,2,4,1,...]
0.66274341934918158097474209710925290 Laplace Limit constant ${\displaystyle \lambda }$ A033259 [0;1,1,1,27,1,1,1,8,2,154,2,4,1,5,...]
0.69314718055994530941723212145817657 Logarithm de 2 ${\displaystyle Ln(2)}$ ${\displaystyle \sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n}}={\frac {1}{1}}-{\frac {1}{2}}+{\frac {1}{3}}-{\frac {1}{4}}+{\frac {1}{5}}-\cdots }$ Sum[n=1 to ∞]{(-1)^(n+1)/n} T A002162 [0;1,2,3,1,6,3,1,1,2,1,1,1,1,3,10,...]
0.78343051071213440705926438652697546 Sophomore's Dream 1 J.Bernoulli ${\displaystyle I_{1}}$ ${\displaystyle \sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n^{n}}}=1-{\frac {1}{2^{2}}}+{\frac {1}{3^{3}}}-{\frac {1}{4^{4}}}+{\frac {1}{5^{5}}}+\dots }$ Sum[ -(-1)^n /n^n] T A083648 [0;1,3,1,1,1,1,1,1,2,4,7,2,1,2,1,1,1,...]
0.78539816339744830961566084581987572 Dirichlet beta(1) ${\displaystyle \beta (1)}$ ${\displaystyle {\frac {\pi }{4}}=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{2n+1}}={\frac {1}{1}}-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+{\frac {1}{9}}-\cdots }$ Sum[n=0 to ∞]{(-1)^n/(2n+1)} T A003881 [0; 1,3,1,1,1,15,2,72,1,9,1,17,1,2,1,5,...]
0.82246703342411321823620758332301259 Traveling Salesman Nielsen-Ramanujan ${\displaystyle {\frac {\zeta (2)}{2}}}$ ${\displaystyle {\frac {\pi ^{2}}{12}}=\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n^{2}}}={\frac {1}{1^{2}}}{-}{\frac {1}{2^{2}}}{+}{\frac {1}{3^{2}}}{-}{\frac {1}{4^{2}}}{+}{\frac {1}{5^{2}}}-\dots }$ Sum[n=1 to ∞]{((-1)^(k+1))/n^2} T A072691 [0;1,4,1,1,1,2,1,1,1,1,3,2,2,4,1,1,1,...]
0.91596559417721901505460351493238411 Catalan constant ${\displaystyle C}$ ${\displaystyle \sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)^{2}}}={\frac {1}{1^{2}}}-{\frac {1}{3^{2}}}+{\frac {1}{5^{2}}}-{\frac {1}{7^{2}}}+\cdots }$ Sum[n=0 to ∞]{(-1)^n/(2n+1)^2} I A006752 [0;1,10,1,8,1,88,4,1,1,7,22,1,2,...]
1.05946309435929526456182529494634170 Ratio of the distance between semi-tones ${\displaystyle {\sqrt[{12}]{2}}}$ ${\displaystyle {\sqrt[{12}]{2}}}$ 2^(1/12) I A010774 [1;16,1,4,2,7,1,1,2,2,7,4,1,2,1,60,1,3,1,2,...]
1,.08232323371113819151600369654116790 Zeta(04) ${\displaystyle \zeta {4}}$ ${\displaystyle {\frac {\pi ^{4}}{90}}=\sum _{n=1}^{\infty }{\frac {1}{n^{4}}}={\frac {1}{1^{4}}}+{\frac {1}{2^{4}}}+{\frac {1}{3^{4}}}+{\frac {1}{4^{4}}}+{\frac {1}{5^{4}}}+\dots }$ Sum[n=1 to ∞]{1/n^4} T A013662 [1;12,6,1,3,1,4,183,1,1,2,1,3,1,1,5,4,2,7,...]
1.1319882487943 ... Viswanaths Archived 2013-04-13 at the Wayback Machine constant ${\displaystyle C_{Vi}}$ ${\displaystyle \lim _{n\to \infty }|a_{n}|^{\frac {1}{n}}}$ A078416 [1;7,1,1,2,1,3,2,1,2,1,8,1,5,1,1,1,9,1,...]
1.20205690315959428539973816151144999 Apéry constant ${\displaystyle \zeta (3)}$ ${\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n^{3}}}={\frac {1}{1^{3}}}+{\frac {1}{2^{3}}}+{\frac {1}{3^{3}}}+{\frac {1}{4^{3}}}+{\frac {1}{5^{3}}}+\cdots \,\!}$ Sum[n=1 to ∞]{1/n^3} I A010774 [1;4,1,18,1,1,1,4,1,9,9,2,1,1,1,2,...]
1.22541670246517764512909830336289053 Gamma(3/4) ${\displaystyle \Gamma ({\tfrac {3}{4}})}$ ${\displaystyle \left(-1+{\frac {3}{4}}\right)!}$ (-1+3/4)! T A068465 [1;4,2,3,2,2,1,1,1,2,1,4,7,1,171,3,2,3,1,1,...]
1.23370055013616982735431137498451889 Favard constant ${\displaystyle {\tfrac {3}{4}}\zeta (2)}$ ${\displaystyle {\frac {\pi ^{2}}{8}}=\sum _{n=0}^{\infty }{\frac {1}{(2n-1)^{2}}}={\frac {1}{1^{2}}}+{\frac {1}{3^{2}}}+{\frac {1}{5^{2}}}+{\frac {1}{7^{2}}}+\dots }$ sum[n=1 to ∞]{1/((2n-1)^2)} T A111003 [1;4,3,1,1,2,2,5,1,1,1,1,2,1,2,1,10,4,3,1,1,...]
1.25992104989487316476721060727822835 Cube root of 2, constante Delian ${\displaystyle {\sqrt[{3}]{2}}}$ ${\displaystyle {\sqrt[{3}]{2}}}$ 2^(1/3) I A002580 [1;3,1,5,1,1,4,1,1,8,1,14,1,10,...]
1.29128599706266354040728259059560054 Sophomore's Dream 2 J.Bernoulli ${\displaystyle I_{2}}$ ${\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n^{n}}}=1+{\frac {1}{2^{2}}}+{\frac {1}{3^{3}}}+{\frac {1}{4^{4}}}+{\frac {1}{5^{5}}}+{\frac {1}{6^{6}}}+\dots }$ Sum[1/(n^n]), {n, 1, ∞}] A073009 [1;3,2,3,4,3,1,2,1,1,6,7,2,5,3,1,2,1,8,1,...]
1.32471795724474602596090885447809734 Plastic number ${\displaystyle \rho }$ ${\displaystyle {\sqrt[{3}]{1+{\sqrt[{3}]{1+{\sqrt[{3}]{1+{\sqrt[{3}]{1+\cdots }}}}}}}}}$ I A060006 [1;3,12,1,1,3,2,3,2,4,2,141,80,2,5,1,2,8,...]
1.41421356237309504880168872420969808 Square root of 2, Pythagoras constant ${\displaystyle {\sqrt {2}}}$ ${\displaystyle \prod _{n=1}^{\infty }1+{\frac {(-1)^{n+1}}{2n-1}}=\left(1{+}{\frac {1}{1}}\right)\left(1{-}{\frac {1}{3}}\right)\left(1{+}{\frac {1}{5}}\right)...}$ prod[n=1 to ∞]{1+(-1)^(n+1)/(2n-1)} I A002193 [1;2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,...]
= [1;(2),...]
1.44466786100976613365833910859643022 Steiner number ${\displaystyle e^{\frac {1}{e}}}$ ${\displaystyle e^{1/e}}$ ... Upper Limit of Tetration A073229 [1;2,4,55,27,1,1,16,9,3,2,8,3,2,1,1,4,1,9,...]
1.53960071783900203869106341467188655 Lieb's Square Ice constant ${\displaystyle W_{2D}}$ ${\displaystyle \lim _{n\to \infty }(f(n))^{n^{-2}}=\left({\frac {4}{3}}\right)^{\frac {3}{2}}}$ (4/3)^(3/2) I A118273 [1;1,1,5,1,4,2,1,6,1,6,1,2,4,1,5,1,1,2,...]
1.57079632679489661923132169163975144 Wallis product ${\displaystyle \pi /2}$ ${\displaystyle \prod _{n=1}^{\infty }\left({\frac {4n^{2}}{4n^{2}-1}}\right)={\frac {2}{1}}\cdot {\frac {2}{3}}\cdot {\frac {4}{3}}\cdot {\frac {4}{5}}\cdot {\frac {6}{5}}\cdot {\frac {6}{7}}\cdot {\frac {8}{7}}\cdot {\frac {8}{9}}\cdots }$ T A019669 [1;1,1,3,31,1,145,1,4,2,8,1,6,1,2,3,1...]
1.60669515241529176378330152319092458 Erdős–Borwein constant ${\displaystyle E_{\,B}}$ ${\displaystyle \sum _{n=1}^{\infty }{\frac {1}{2^{n}-1}}={\frac {1}{1}}+{\frac {1}{3}}+{\frac {1}{7}}+{\frac {1}{15}}+\cdots \,\!}$ sum[n=1 to ∞]{1/(2^n-1)} I A065442 [1;1,1,1,1,5,2,1,2,29,4,1,2,2,2,2,6,1,7,1,...]
1.61803398874989484820458633436563812 Phi, Golden ratio ${\displaystyle \varphi }$ ${\displaystyle {\frac {1+{\sqrt {5}}}{2}}={\sqrt {1+{\sqrt {1+{\sqrt {1+{\sqrt {1+\cdots }}}}}}}}}$ (1+5^(1/2))/2 I A001622 [0;1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,...]
= [0;(1),...]
1.64493406684822643647241516664602519 Zeta(2) ${\displaystyle \zeta (\,2)}$ ${\displaystyle {\frac {\pi ^{2}}{6}}=\sum _{n=1}^{\infty }{\frac {1}{n^{2}}}={\frac {1}{1^{2}}}+{\frac {1}{2^{2}}}+{\frac {1}{3^{2}}}+{\frac {1}{4^{2}}}+\cdots }$ Sum[n=1 to ∞]{1/n^2} T A013661 [1;1,1,1,4,2,4,7,1,4,2,3,4,10 1,2,1,1,1,15,...]
1.66168794963359412129581892274995074 Somos' quadratic recurrence constant ${\displaystyle \sigma }$ ${\displaystyle {\sqrt {1{\sqrt {2{\sqrt {3\cdots }}}}}}=1^{1/2};2^{1/4};3^{1/8}\cdots }$ T A065481 [1;1,1,1,21,1,1,1,6,4,2,1,1,2,1,3,1,13,13,...]
1.73205080756887729352744634150587237 Theodorus constant ${\displaystyle {\sqrt {3}}}$ ${\displaystyle {\sqrt {3}}}$ 3^(1/2) I A002194 [1;1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,...]
= [1;(1,2),...]
1.75793275661800453270881963821813852 Kasner number ${\displaystyle R}$ ${\displaystyle {\sqrt {1+{\sqrt {2+{\sqrt {3+{\sqrt {4+\cdots }}}}}}}}}$ A072449 [1;1,3,7,1,1,1,2,3,1,4,1,1,2,1,2,20,1,2,2,...]
1.77245385090551602729816748334114518 Carlson-Levin constant ${\displaystyle \Gamma ({\tfrac {1}{2}})}$ ${\displaystyle {\sqrt {\pi }}=\left(-{\frac {1}{2}}\right)!}$ sqrt (pi) T A002161 [1;1,3,2,1,1,6,1,28,13,1,1,2,18,1,1,1,83,1,...]
2.29558714939263807403429804918949038 P, Universal parabolic constant ${\displaystyle P}$ ${\displaystyle \ln(1+{\sqrt {2}})+{\sqrt {2}}}$ ln(1+sqrt 2)+sqrt 2 T A103710 [2;3,2,1,1,1,1,3,3,1,1,4,2,3,2,7,1,6,1,8,...]
2.30277563773199464655961063373524797 Bronze Number ${\displaystyle \sigma _{\,Rr}}$ ${\displaystyle {\frac {3+{\sqrt {13}}}{2}}=1+{\sqrt {3+{\sqrt {3+{\sqrt {3+{\sqrt {3+\cdots }}}}}}}}}$ (3+sqrt 13)/2 I A098316 [3;3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,...]
= [3;(3),...]
2.37313822083125090564344595189447424 Lévy constant2 ${\displaystyle 2\,\ln \,\gamma }$ ${\displaystyle {\frac {\pi ^{2}}{6\ln(2)}}}$ Pi^(2)/(6*ln(2)) T A174606 [2;2,1,2,8,57,9,32,1,1,2,1,2,1,2,1,2,1,3,2,...]
2.50662827463100050241576528481104525 square root of 2 pi ${\displaystyle {\sqrt {2\pi }}}$ ${\displaystyle {\sqrt {2\pi }}=\lim _{n\to \infty }{\frac {n!\;e^{n}}{n^{n}{\sqrt {n}}}}}$ sqrt (2*pi) T A019727 [2;1,1,37,4,1,1,1,1,9,1,1,2,8,6,1,2,2,1,3,...]
2.66514414269022518865029724987313985 Gelfond-Schneider constant ${\displaystyle G_{_{\,GS}}}$ ${\displaystyle 2^{\sqrt {2}}}$ 2^sqrt{2} T A007507 [2;1,1,1,72,3,4,1,3,2,1,1,1,14,1,2,1,1,3,1,...]
2.68545200106530644530971483548179569 Khintchin constant ${\displaystyle K_{\,0}}$ ${\displaystyle \prod _{n=1}^{\infty }\left[{1+{1 \over n(n+2)}}\right]^{\ln n/\ln 2}}$ prod[n=1 to ∞]{(1+1/(n(n+2)))^((ln(n)/ln(2))} ? A002210 [2;1,2,5,1,1,2,1,1,3,10,2,1,3,2,24,1,3,2,...]
3.27582291872181115978768188245384386 Khinchin-Lévy constant ${\displaystyle \gamma }$ ${\displaystyle e^{\pi ^{2}/(12\ln 2)}}$ e^(\pi^2/(12 ln(2)) A086702 [3;3,1,1,1,2,29,1,130,1,12,3,8,2,4,1,3,55,...]
3.35988566624317755317201130291892717 Reciprocal Fibonacci constant ${\displaystyle \Psi }$ ${\displaystyle \sum _{n=1}^{\infty }{\frac {1}{F_{n}}}={\frac {1}{1}}+{\frac {1}{1}}+{\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{5}}+{\frac {1}{8}}+{\frac {1}{13}}+\cdots }$ A079586 [3;2,1,3,1,1,13,2,3,3,2,1,1,6,3,2,4,362,...]
4.13273135412249293846939188429985264 Root of 2 e pi ${\displaystyle {\sqrt {2e\pi }}}$ ${\displaystyle {\sqrt {2e\pi }}}$ sqrt(2e pi) T A019633 [4;7,1,1,6,1,5,1,1,1,8,3,1,2,2,15,2,1,1,2,4,...]
6.58088599101792097085154240388648649 Froda constant ${\displaystyle 2^{\,e}}$ ${\displaystyle 2^{e}}$ 2^e [6;1,1,2,1,1,2,3,1,14,11,4,3,1,1,7,5,5,2,7,...]
9.86960440108935861883449099987615114 Pi Squared ${\displaystyle \pi ^{2}}$ ${\displaystyle 6\sum _{n=1}^{\infty }{\frac {1}{n^{2}}}={\frac {6}{1^{2}}}+{\frac {6}{2^{2}}}+{\frac {6}{3^{2}}}+{\frac {6}{4^{2}}}+\cdots }$ 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,...]
23.1406926327792690057290863679485474 Gelfond constant ${\displaystyle e^{\pi }}$ ${\displaystyle \sum _{n=0}^{\infty }{\frac {\pi ^{n}}{n!}}={\frac {\pi ^{1}}{1}}+{\frac {\pi ^{2}}{2!}}+{\frac {\pi ^{3}}{3!}}+{\frac {\pi ^{4}}{4!}}+\cdots }$ Sum[n=0 to ∞]{(pi^n)/n!} T A039661 [23;7,9,3,1,1,591,2,9,1,2,34,1,16,1,30,1,...]

References

1. "Compendium of Mathematical Symbols". Math Vault. 2020-03-01. Retrieved 2020-08-26.
2. "Constant | mathematics and logic". Encyclopedia Britannica. Retrieved 2020-08-26.
3. Weisstein, Eric W. "Constant". mathworld.wolfram.com. Retrieved 2020-08-26.

Books

• Daniel Zwillinger (2012). Standard Mathematical Tables and Formulae. Imperial College Press. ISBN 978-1-4398-3548-7.
• Eric W. Weisstein (2003). CRC Concise Encyclopedia of Mathematics. Chapman & Hall/CRC. ISBN 1-58488-347-2.
• Lloyd Kilford (2008). Modular Forms, a Classical and Computational Introduction. Imperial College Press. ISBN 978-1-84816-213-6.