Value
Name
Symbol LaTeX
Formula
Type OEIS
Continued fraction
3.24697960371746706105000976800847962
Silver, Tutte–Beraha constant
ς
{\displaystyle \varsigma }
2
+
2
cos
(
2
π
/
7
)
=
2
+
2
+
7
+
7
7
+
7
7
+
⋯
3
3
3
1
+
7
+
7
7
+
7
7
+
⋯
3
3
3
{\displaystyle 2+2\cos(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)
I 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
C
P
a
{\displaystyle C_{Pa}}
∏
n
=
2
∞
2
φ
φ
+
φ
n
,
φ
=
F
i
{\displaystyle \prod _{n=2}^{\infty }{\frac {2\varphi }{\varphi +\varphi _{n}}}\;,\varphi ={Fi}}
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
R
5
{\displaystyle R_{5}}
5
+
5
+
5
−
5
+
5
+
5
+
5
−
⋯
=
2
+
5
+
15
−
6
5
2
{\displaystyle \scriptstyle {\sqrt {5+{\sqrt {5+{\sqrt {5-{\sqrt {5+{\sqrt {5+{\sqrt {5+{\sqrt {5-\cdots }}}}}}}}}}}}}}\;=\textstyle {\frac {2+{\sqrt {5}}+{\sqrt {15-6{\sqrt {5}}}}}{2}}}
(2+sqrt(5)+sqrt(15-6 sqrt(5)))/2
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
5
{\displaystyle {\sqrt {5}}}
∀
n
=
5
,
∑
k
=
0
n
−
1
e
2
k
2
π
i
n
=
1
+
e
2
π
i
5
+
e
8
π
i
5
+
e
18
π
i
5
+
e
32
π
i
5
{\displaystyle \scriptstyle \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,...]
3.62560990822190831193068515586767200
Gamma(1/4)
Γ
(
1
4
)
{\displaystyle \Gamma ({\tfrac {1}{4}})}
4
(
1
4
)
!
=
(
−
3
4
)
!
{\displaystyle 4\left({\frac {1}{4}}\right)!=\left(-{\frac {3}{4}}\right)!}
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
C
M
R
B
{\displaystyle C_{_{MRB}}}
∑
n
=
1
∞
(
−
1
)
n
(
n
1
/
n
−
1
)
=
−
1
1
+
2
2
−
3
3
+
4
4
…
{\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
ρ
{\displaystyle {\rho }}
∏
n
=
3
∞
cos
(
π
n
)
=
cos
(
π
3
)
cos
(
π
4
)
cos
(
π
5
)
…
{\displaystyle \prod _{n=3}^{\infty }\cos \left({\frac {\pi }{n}}\right)=\cos \left({\frac {\pi }{3}}\right)\cos \left({\frac {\pi }{4}}\right)\cos \left({\frac {\pi }{5}}\right)\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)
e
γ
{\displaystyle e^{\gamma }}
∏
n
=
1
∞
e
1
n
1
+
1
n
=
∏
n
=
0
∞
(
∏
k
=
0
n
(
k
+
1
)
(
−
1
)
k
+
1
(
n
k
)
)
1
n
+
1
=
{\displaystyle \prod _{n=1}^{\infty }{\frac {e^{\frac {1}{n}}}{1+{\tfrac {1}{n}}}}=\prod _{n=0}^{\infty }\left(\prod _{k=0}^{n}(k+1)^{(-1)^{k+1}{n \choose k}}\right)^{\frac {1}{n+1}}=}
(
2
1
)
1
/
2
(
2
2
1
⋅
3
)
1
/
3
(
2
3
⋅
4
1
⋅
3
3
)
1
/
4
(
2
4
⋅
4
4
1
⋅
3
6
⋅
5
)
1
/
5
…
{\displaystyle \textstyle \left({\frac {2}{1}}\right)^{1/2}\left({\frac {2^{2}}{1\cdot 3}}\right)^{1/3}\left({\frac {2^{3}\cdot 4}{1\cdot 3^{3}}}\right)^{1/4}\left({\frac {2^{4}\cdot 4^{4}}{1\cdot 3^{6}\cdot 5}}\right)^{1/5}\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
A
{\displaystyle {A}}
e
1
12
−
ζ
′
(
−
1
)
=
e
1
8
−
1
2
∑
n
=
0
∞
1
n
+
1
∑
k
=
0
n
(
−
1
)
k
(
n
k
)
(
k
+
1
)
2
ln
(
k
+
1
)
{\displaystyle e^{{\frac {1}{12}}-\zeta ^{\prime }(-1)}=e^{{\frac {1}{8}}-{\frac {1}{2}}\sum \limits _{n=0}^{\infty }{\frac {1}{n+1}}\sum \limits _{k=0}^{n}\left(-1\right)^{k}{\binom {n}{k}}\left(k+1\right)^{2}\ln(k+1)}}
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
e
2
{\displaystyle e^{2}}
∑
n
=
0
∞
2
n
n
!
=
1
+
2
+
2
2
2
!
+
2
3
3
!
+
2
4
4
!
+
2
5
5
!
+
…
{\displaystyle \sum _{n=0}^{\infty }{\frac {2^{n}}{n!}}=1+2+{\frac {2^{2}}{2!}}+{\frac {2^{3}}{3!}}+{\frac {2^{4}}{4!}}+{\frac {2^{5}}{5!}}+\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,...]
1.01494160640965362502120255427452028
Gieseking constant
G
G
i
{\displaystyle {G_{Gi}}}
3
3
4
(
1
−
∑
n
=
0
∞
1
(
3
n
+
2
)
2
+
∑
n
=
1
∞
1
(
3
n
+
1
)
2
)
=
{\displaystyle {\frac {3{\sqrt {3}}}{4}}\left(1-\sum _{n=0}^{\infty }{\frac {1}{(3n+2)^{2}}}+\sum _{n=1}^{\infty }{\frac {1}{(3n+1)^{2}}}\right)=}
3
3
4
(
1
−
1
2
2
+
1
4
2
−
1
5
2
+
1
7
2
−
1
8
2
+
1
10
2
±
…
)
{\displaystyle \textstyle {\frac {3{\sqrt {3}}}{4}}\left(1-{\frac {1}{2^{2}}}+{\frac {1}{4^{2}}}-{\frac {1}{5^{2}}}+{\frac {1}{7^{2}}}-{\frac {1}{8^{2}}}+{\frac {1}{10^{2}}}\pm \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
ϖ
{\displaystyle {\varpi }}
π
G
=
4
2
π
(
1
4
!
)
2
{\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
G
{\displaystyle {G}}
1
a
g
m
(
1
,
2
)
=
4
2
(
1
4
!
)
2
π
3
/
2
A
g
m
:
A
r
i
t
h
m
e
t
i
c
−
g
e
o
m
e
t
r
i
c
m
e
a
n
{\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)
ζ
(
6
)
{\displaystyle \zeta (6)}
π
6
945
=
∏
n
=
1
∞
1
1
−
p
n
−
6
p
n
:
p
r
i
m
o
=
1
1
−
2
−
6
⋅
1
1
−
3
−
6
⋅
1
1
−
5
−
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
1
ζ
(
2
)
{\displaystyle {\frac {1}{\zeta (2)}}}
6
π
2
=
∏
n
=
0
∞
(
1
−
1
p
n
2
)
p
n
:
p
r
i
m
o
=
(
1
−
1
2
2
)
(
1
−
1
3
2
)
(
1
−
1
5
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
π
2
2
{\displaystyle {\frac {\pi }{2{\sqrt {2}}}}}
∑
n
=
1
∞
(
−
1
)
⌊
n
−
1
2
⌋
2
n
+
1
=
1
1
+
1
3
−
1
5
−
1
7
+
1
9
+
1
11
−
…
{\displaystyle \sum _{n=1}^{\infty }{\frac {(-1)^{\lfloor {\frac {n-1}{2}}\rfloor }}{2n+1}}={\frac {1}{1}}+{\frac {1}{3}}-{\frac {1}{5}}-{\frac {1}{7}}+{\frac {1}{9}}+{\frac {1}{11}}-\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
F
{\displaystyle {F}}
∫
0
∞
1
Γ
(
x
)
d
x
.
=
e
+
∫
0
∞
e
−
x
π
2
+
ln
2
x
d
x
{\displaystyle \int _{0}^{\infty }{\frac {1}{\Gamma (x)}}\,dx.=e+\int _{0}^{\infty }{\frac {e^{-x}}{\pi ^{2}+\ln ^{2}x}}\,dx}
N[int[0 to ∞] {1/Gamma(x)}]
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
e
{\displaystyle {\sqrt {e}}}
∑
n
=
0
∞
1
2
n
n
!
=
∑
n
=
0
∞
1
(
2
n
)
!
!
=
1
1
+
1
2
+
1
8
+
1
48
+
⋯
{\displaystyle \sum _{n=0}^{\infty }{\frac {1}{2^{n}n!}}=\sum _{n=0}^{\infty }{\frac {1}{(2n)!!}}={\frac {1}{1}}+{\frac {1}{2}}+{\frac {1}{8}}+{\frac {1}{48}}+\cdots }
sum[n=0 to ∞]{1/(2^n n!)}
T A019774
[1;1,1,1,5,1,1,9,1,1,13,1,1,17,1,1,21,1,1,...]
i
i, imaginary unit
i
{\displaystyle {i}}
−
1
=
ln
(
−
1
)
π
e
i
π
=
−
1
{\displaystyle {\sqrt {-1}}={\frac {\ln(-1)}{\pi }}\qquad \qquad \mathrm {e} ^{i\,\pi }=-1}
sqrt(-1)
C
262537412640768743.999999999999250073
Hermite-Ramanujan constant
R
{\displaystyle {R}}
e
π
163
{\displaystyle e^{\pi {\sqrt {163}}}}
e^(π sqrt(163))
T A060295
[262537412640768743;1,1333462407511,1,8,1,1,5,...]
4.81047738096535165547303566670383313
John constant
γ
{\displaystyle \gamma }
i
i
=
i
−
i
=
i
1
i
=
(
i
i
)
−
1
=
e
π
2
{\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
α
{\displaystyle \alpha }
π
l
n
(
2
)
=
∑
n
=
0
∞
4
(
−
1
)
n
2
n
+
1
∑
n
=
1
∞
(
−
1
)
n
+
1
n
=
4
1
−
4
3
+
4
5
−
4
7
+
4
9
−
…
1
1
−
1
2
+
1
3
−
1
4
+
1
5
−
…
{\displaystyle {\frac {\pi }{ln(2)}}={\frac {\sum _{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)
t
h
1
{\displaystyle th\,1}
e
−
1
e
e
+
1
e
=
e
2
−
1
e
2
+
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,...]
0.69777465796400798200679059255175260
Continued Fraction constant
C
C
F
{\displaystyle {C}_{CF}}
J
1
(
2
)
J
0
(
2
)
F
u
n
c
t
i
o
n
J
k
(
)
B
e
s
s
e
l
=
∑
n
=
0
∞
n
n
!
n
!
∑
n
=
0
∞
1
n
!
n
!
=
0
1
+
1
1
+
2
4
+
3
36
+
4
576
+
…
1
1
+
1
1
+
1
4
+
1
36
+
1
576
+
…
{\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,...]
0.36787944117144232159552377016146086
Inverse Napier constant
1
e
{\displaystyle {\frac {1}{e}}}
∑
n
=
0
∞
(
−
1
)
n
n
!
=
1
0
!
−
1
1
!
+
1
2
!
−
1
3
!
+
1
4
!
−
1
5
!
+
…
{\displaystyle \sum _{n=0}^{\infty }{\frac {(-1)^{n}}{n!}}={\frac {1}{0!}}-{\frac {1}{1!}}+{\frac {1}{2!}}-{\frac {1}{3!}}+{\frac {1}{4!}}-{\frac {1}{5!}}+\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,...]
2.71828182845904523536028747135266250
Napier constant
e
{\displaystyle e}
∑
n
=
0
∞
1
n
!
=
1
0
!
+
1
1
+
1
2
!
+
1
3
!
+
1
4
!
+
1
5
!
+
⋯
{\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,...]
0.49801566811835604271369111746219809 - 0.15494982830181068512495513048388 i
Factorial of i
i
!
{\displaystyle i\,!}
Γ
(
1
+
i
)
=
i
Γ
(
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,...] i
0.43828293672703211162697516355126482 + 0.36059247187138548595294052690600 i
Infinite Tetration of i
∞
i
{\displaystyle {}^{\infty }i}
lim
n
→
∞
n
i
=
lim
n
→
∞
i
i
⋅
⋅
i
⏟
n
{\displaystyle \lim _{n\to \infty }{}^{n}i=\lim _{n\to \infty }\underbrace {i^{i^{\cdot ^{\cdot ^{i}}}}} _{n}}
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, ...] i
0.56755516330695782538461314419245334
Module of Tetration of i
|
∞
i
|
{\displaystyle |{}^{\infty }i|}
lim
n
→
∞
|
n
i
|
=
|
lim
n
→
∞
i
i
⋅
⋅
i
⏟
n
|
{\displaystyle \lim _{n\to \infty }\left|{}^{n}i\right|=\left|\lim _{n\to \infty }\underbrace {i^{i^{\cdot ^{\cdot ^{i}}}}} _{n}\right|}
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
M
{\displaystyle M}
lim
n
→
∞
(
∑
p
≤
n
1
p
−
ln
(
ln
(
n
)
)
)
{\displaystyle \lim _{n\rightarrow \infty }\left(\sum _{p\leq n}{\frac {1}{p}}-\ln(\ln(n))\right)}
A077761
[0;3,1,4,1,2,5,2,1,1,1,1,13,4,2,4,2,1,33,...]
1.9287800...
Wright constant
ω
{\displaystyle \omega }
⌊
2
2
2
⋅
⋅
2
ω
⌋
{\displaystyle \left\lfloor 2^{2^{2^{\cdot ^{\cdot ^{2^{\omega }}}}}}\right\rfloor }
{\displaystyle \quad }
⌊
2
ω
⌋
{\displaystyle \left\lfloor 2^{\omega }\right\rfloor }
⌊
2
2
ω
⌋
{\displaystyle \left\lfloor 2^{2^{\omega }}\right\rfloor }
⌊
2
2
2
ω
⌋
{\displaystyle \left\lfloor 2^{2^{2^{\omega }}}\right\rfloor }
…
{\displaystyle \dots }
A086238
[1; 1, 13, 24, 2, 1, 1, 3, 1, 1, 3]
0.37395581361920228805472805434641641
Artin constant
C
A
r
t
i
n
{\displaystyle C_{Artin}}
∏
n
=
1
∞
(
1
−
1
p
n
(
p
n
−
1
)
)
{\displaystyle \prod _{n=1}^{\infty }\left(1-{\frac {1}{p_{n}(p_{n}-1)}}\right)}
n : 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
δ
{\displaystyle {\delta }}
lim
n
→
∞
x
n
+
1
−
x
n
x
n
+
2
−
x
n
+
1
x
∈
(
3
,
8284
;
3
,
8495
)
{\displaystyle \lim _{n\to \infty }{\frac {x_{n+1}-x_{n}}{x_{n+2}-x_{n+1}}}\qquad \scriptstyle x\in (3,8284;\,3,8495)}
x
n
+
1
=
a
x
n
(
1
−
x
n
)
o
x
n
+
1
=
a
sin
(
x
n
)
{\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
α
{\displaystyle \alpha }
lim
n
→
∞
d
n
d
n
+
1
{\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
H
2
(
2
)
{\displaystyle H_{2}(2)}
π
ln
(
3
)
3
{\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)
β
(
3
)
{\displaystyle \beta (3)}
π
3
32
=
∑
n
=
1
∞
−
1
n
+
1
(
−
1
+
2
n
)
3
=
1
1
3
−
1
3
3
+
1
5
3
−
1
7
3
+
…
{\displaystyle {\frac {\pi ^{3}}{32}}=\sum _{n=1}^{\infty }{\frac {-1^{n+1}}{(-1+2n)^{3}}}={\frac {1}{1^{3}}}{-}{\frac {1}{3^{3}}}{+}{\frac {1}{5^{3}}}{-}{\frac {1}{7^{3}}}{+}\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
B
2
{\displaystyle B_{\,2}}
∑
(
1
p
+
1
p
+
2
)
p
,
p
+
2
:
p
r
i
m
o
s
=
(
1
3
+
1
5
)
+
(
1
5
+
1
7
)
+
(
1
11
+
1
13
)
+
…
{\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
B
4
{\displaystyle B_{\,4}}
(
1
5
+
1
7
+
1
11
+
1
13
)
p
,
p
+
2
,
p
+
4
,
p
+
6
:
p
r
i
m
e
s
+
(
1
11
+
1
13
+
1
17
+
1
19
)
+
…
{\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
π
e
{\displaystyle \pi ^{e}}
π
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
π
{\displaystyle \pi }
lim
n
→
∞
2
n
2
−
2
+
2
+
⋯
+
2
⏟
n
{\displaystyle \lim _{n\to \infty }\,2^{n}\underbrace {\sqrt {2-{\sqrt {2+{\sqrt {2+\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
e
−
e
{\displaystyle e^{-e}}
e
−
e
{\displaystyle e^{-e}}
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
i
i
{\displaystyle i^{i}}
e
−
π
2
{\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
β
{\displaystyle \beta }
1
2
π
{\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
Q
{\displaystyle Q}
∏
n
=
1
∞
(
1
−
1
2
n
)
=
(
1
−
1
2
1
)
(
1
−
1
2
2
)
(
1
−
1
2
3
)
…
{\displaystyle \prod _{n=1}^{\infty }\left(1-{\frac {1}{2^{n}}}\right)=\left(1{-}{\frac {1}{2^{1}}}\right)\left(1{-}{\frac {1}{2^{2}}}\right)\left(1{-}{\frac {1}{2^{3}}}\right)\dots }
prod[n=1 to ∞]{1-1/2^n}
A048651
0.31830988618379067153776752674502872
Inverse of Pi , Ramanujan
1
π
{\displaystyle {\frac {1}{\pi }}}
2
2
9801
∑
n
=
0
∞
(
4
n
)
!
(
1103
+
26390
n
)
(
n
!
)
4
396
4
n
{\displaystyle {\frac {2{\sqrt {2}}}{9801}}\sum _{n=0}^{\infty }{\frac {(4n)!(1103+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
W
W
E
{\displaystyle W_{_{WE}}}
e
π
8
π
4
∗
2
3
/
4
(
1
4
!
)
2
{\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
Ω
{\displaystyle \Omega }
W
(
1
)
=
∑
n
=
1
∞
(
−
n
)
n
−
1
n
!
=
1
−
1
+
3
2
−
8
3
+
125
24
−
…
{\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
γ
{\displaystyle \gamma }
−
ψ
(
1
)
=
∑
n
=
1
∞
∑
k
=
0
∞
(
−
1
)
k
2
n
+
k
{\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
π
3
3
{\displaystyle {\frac {\pi }{3{\sqrt {3}}}}}
∑
n
=
1
∞
1
n
(
2
n
n
)
=
1
−
1
2
+
1
4
−
1
5
+
1
7
−
1
8
+
⋯
{\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n{2n \choose n}}}=1-{\frac {1}{2}}+{\frac {1}{4}}-{\frac {1}{5}}+{\frac {1}{7}}-{\frac {1}{8}}+\cdots }
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
2
π
{\displaystyle {\frac {2}{\pi }}}
2
2
⋅
2
+
2
2
⋅
2
+
2
+
2
2
⋯
{\displaystyle {\frac {\sqrt {2}}{2}}\cdot {\frac {\sqrt {2+{\sqrt {2}}}}{2}}\cdot {\frac {\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}{2}}\cdots }
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
C
2
{\displaystyle C_{2}}
∏
p
=
3
∞
p
(
p
−
2
)
(
p
−
1
)
2
{\displaystyle \prod _{p=3}^{\infty }{\frac {p(p-2)}{(p-1)^{2}}}}
prod[p=3 to ∞]{p(p-2)/(p-1)^2
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
λ
{\displaystyle \lambda }
A033259
[0;1,1,1,27,1,1,1,8,2,154,2,4,1,5,...]
0.69314718055994530941723212145817657
Logarithm de 2
L
n
(
2
)
{\displaystyle Ln(2)}
∑
n
=
1
∞
(
−
1
)
n
+
1
n
=
1
1
−
1
2
+
1
3
−
1
4
+
1
5
−
⋯
{\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
I
1
{\displaystyle I_{1}}
∑
n
=
1
∞
(
−
1
)
n
+
1
n
n
=
1
−
1
2
2
+
1
3
3
−
1
4
4
+
1
5
5
+
…
{\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)
β
(
1
)
{\displaystyle \beta (1)}
π
4
=
∑
n
=
0
∞
(
−
1
)
n
2
n
+
1
=
1
1
−
1
3
+
1
5
−
1
7
+
1
9
−
⋯
{\displaystyle {\frac {\pi }{4}}=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{2n+1}}={\frac {1}{1}}-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+{\frac {1}{9}}-\cdots }
Sum[n=0 to ∞]{(-1)^n/(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
ζ
(
2
)
2
{\displaystyle {\frac {\zeta (2)}{2}}}
π
2
12
=
∑
n
=
1
∞
(
−
1
)
n
+
1
n
2
=
1
1
2
−
1
2
2
+
1
3
2
−
1
4
2
+
1
5
2
−
…
{\displaystyle {\frac {\pi ^{2}}{12}}=\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n^{2}}}={\frac {1}{1^{2}}}{-}{\frac {1}{2^{2}}}{+}{\frac {1}{3^{2}}}{-}{\frac {1}{4^{2}}}{+}{\frac {1}{5^{2}}}-\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
C
{\displaystyle C}
∑
n
=
0
∞
(
−
1
)
n
(
2
n
+
1
)
2
=
1
1
2
−
1
3
2
+
1
5
2
−
1
7
2
+
⋯
{\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
2
12
{\displaystyle {\sqrt[{12}]{2}}}
2
12
{\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)
ζ
4
{\displaystyle \zeta {4}}
π
4
90
=
∑
n
=
1
∞
1
n
4
=
1
1
4
+
1
2
4
+
1
3
4
+
1
4
4
+
1
5
4
+
…
{\displaystyle {\frac {\pi ^{4}}{90}}=\sum _{n=1}^{\infty }{\frac {1}{n^{4}}}={\frac {1}{1^{4}}}+{\frac {1}{2^{4}}}+{\frac {1}{3^{4}}}+{\frac {1}{4^{4}}}+{\frac {1}{5^{4}}}+\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
C
V
i
{\displaystyle C_{Vi}}
lim
n
→
∞
|
a
n
|
1
n
{\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
ζ
(
3
)
{\displaystyle \zeta (3)}
∑
n
=
1
∞
1
n
3
=
1
1
3
+
1
2
3
+
1
3
3
+
1
4
3
+
1
5
3
+
⋯
{\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n^{3}}}={\frac {1}{1^{3}}}+{\frac {1}{2^{3}}}+{\frac {1}{3^{3}}}+{\frac {1}{4^{3}}}+{\frac {1}{5^{3}}}+\cdots \,\!}
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)
Γ
(
3
4
)
{\displaystyle \Gamma ({\tfrac {3}{4}})}
(
−
1
+
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
3
4
ζ
(
2
)
{\displaystyle {\tfrac {3}{4}}\zeta (2)}
π
2
8
=
∑
n
=
0
∞
1
(
2
n
−
1
)
2
=
1
1
2
+
1
3
2
+
1
5
2
+
1
7
2
+
…
{\displaystyle {\frac {\pi ^{2}}{8}}=\sum _{n=0}^{\infty }{\frac {1}{(2n-1)^{2}}}={\frac {1}{1^{2}}}+{\frac {1}{3^{2}}}+{\frac {1}{5^{2}}}+{\frac {1}{7^{2}}}+\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
2
3
{\displaystyle {\sqrt[{3}]{2}}}
2
3
{\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
I
2
{\displaystyle I_{2}}
∑
n
=
1
∞
1
n
n
=
1
+
1
2
2
+
1
3
3
+
1
4
4
+
1
5
5
+
1
6
6
+
…
{\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
ρ
{\displaystyle \rho }
1
+
1
+
1
+
1
+
⋯
3
3
3
3
{\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
2
{\displaystyle {\sqrt {2}}}
∏
n
=
1
∞
1
+
(
−
1
)
n
+
1
2
n
−
1
=
(
1
+
1
1
)
(
1
−
1
3
)
(
1
+
1
5
)
.
.
.
{\displaystyle \prod _{n=1}^{\infty }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,...]
1.44466786100976613365833910859643022
Steiner number
e
1
e
{\displaystyle e^{\frac {1}{e}}}
e
1
/
e
{\displaystyle e^{1/e}}
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
W
2
D
{\displaystyle W_{2D}}
lim
n
→
∞
(
f
(
n
)
)
n
−
2
=
(
4
3
)
3
2
{\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
π
/
2
{\displaystyle \pi /2}
∏
n
=
1
∞
(
4
n
2
4
n
2
−
1
)
=
2
1
⋅
2
3
⋅
4
3
⋅
4
5
⋅
6
5
⋅
6
7
⋅
8
7
⋅
8
9
⋯
{\displaystyle \prod _{n=1}^{\infty }\left({\frac {4n^{2}}{4n^{2}-1}}\right)={\frac {2}{1}}\cdot {\frac {2}{3}}\cdot {\frac {4}{3}}\cdot {\frac {4}{5}}\cdot {\frac {6}{5}}\cdot {\frac {6}{7}}\cdot {\frac {8}{7}}\cdot {\frac {8}{9}}\cdots }
T A019669
[1;1,1,3,31,1,145,1,4,2,8,1,6,1,2,3,1...]
1.60669515241529176378330152319092458
Erdős–Borwein constant
E
B
{\displaystyle E_{\,B}}
∑
n
=
1
∞
1
2
n
−
1
=
1
1
+
1
3
+
1
7
+
1
15
+
⋯
{\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
φ
{\displaystyle \varphi }
1
+
5
2
=
1
+
1
+
1
+
1
+
⋯
{\displaystyle {\frac {1+{\sqrt {5}}}{2}}={\sqrt {1+{\sqrt {1+{\sqrt {1+{\sqrt {1+\cdots }}}}}}}}}
(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,...]
1.64493406684822643647241516664602519
Zeta(2)
ζ
(
2
)
{\displaystyle \zeta (\,2)}
π
2
6
=
∑
n
=
1
∞
1
n
2
=
1
1
2
+
1
2
2
+
1
3
2
+
1
4
2
+
⋯
{\displaystyle {\frac {\pi ^{2}}{6}}=\sum _{n=1}^{\infty }{\frac {1}{n^{2}}}={\frac {1}{1^{2}}}+{\frac {1}{2^{2}}}+{\frac {1}{3^{2}}}+{\frac {1}{4^{2}}}+\cdots }
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
σ
{\displaystyle \sigma }
1
2
3
⋯
=
1
1
/
2
;
2
1
/
4
;
3
1
/
8
⋯
{\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
3
{\displaystyle {\sqrt {3}}}
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,...]
1.75793275661800453270881963821813852
Kasner number
R
{\displaystyle R}
1
+
2
+
3
+
4
+
⋯
{\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
Γ
(
1
2
)
{\displaystyle \Gamma ({\tfrac {1}{2}})}
π
=
(
−
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
P
{\displaystyle P}
ln
(
1
+
2
)
+
2
{\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
σ
R
r
{\displaystyle \sigma _{\,Rr}}
3
+
13
2
=
1
+
3
+
3
+
3
+
3
+
⋯
{\displaystyle {\frac {3+{\sqrt {13}}}{2}}=1+{\sqrt {3+{\sqrt {3+{\sqrt {3+{\sqrt {3+\cdots }}}}}}}}}
(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,...]
2.37313822083125090564344595189447424
Lévy constant2
2
ln
γ
{\displaystyle 2\,\ln \,\gamma }
π
2
6
ln
(
2
)
{\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
2
π
{\displaystyle {\sqrt {2\pi }}}
2
π
=
lim
n
→
∞
n
!
e
n
n
n
n
{\displaystyle {\sqrt {2\pi }}=\lim _{n\to \infty }{\frac {n!\;e^{n}}{n^{n}{\sqrt {n}}}}}
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
G
G
S
{\displaystyle G_{_{\,GS}}}
2
2
{\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
K
0
{\displaystyle K_{\,0}}
∏
n
=
1
∞
[
1
+
1
n
(
n
+
2
)
]
ln
n
/
ln
2
{\displaystyle \prod _{n=1}^{\infty }\left[{1+{1 \over n(n+2)}}\right]^{\ln n/\ln 2}}
prod[n=1 to ∞]{(1+1/(n(n+2)))^((ln(n)/ln(2))}
?
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
γ
{\displaystyle \gamma }
e
π
2
/
(
12
ln
2
)
{\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
Ψ
{\displaystyle \Psi }
∑
n
=
1
∞
1
F
n
=
1
1
+
1
1
+
1
2
+
1
3
+
1
5
+
1
8
+
1
13
+
⋯
{\displaystyle \sum _{n=1}^{\infty }{\frac {1}{F_{n}}}={\frac {1}{1}}+{\frac {1}{1}}+{\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{5}}+{\frac {1}{8}}+{\frac {1}{13}}+\cdots }
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
2
e
π
{\displaystyle {\sqrt {2e\pi }}}
2
e
π
{\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
2
e
{\displaystyle 2^{\,e}}
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
π
2
{\displaystyle \pi ^{2}}
6
∑
n
=
1
∞
1
n
2
=
6
1
2
+
6
2
2
+
6
3
2
+
6
4
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
e
π
{\displaystyle e^{\pi }}
∑
n
=
0
∞
π
n
n
!
=
π
1
1
+
π
2
2
!
+
π
3
3
!
+
π
4
4
!
+
⋯
{\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,...]
![{\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 }}}}}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/63c2ba5c39dd844946fe3ac7702fa5e6b6460472)
![{\displaystyle \sum _{n=1}^{\infty }({-}1)^{n}(n^{1/n}{-}1)=-{\sqrt[{1}]{1}}+{\sqrt[{2}]{2}}-{\sqrt[{3}]{3}}+{\sqrt[{4}]{4}}\,\dots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/870bc7fa0415cfa4f3c3fb9253254c65e8e9d967)
![{\displaystyle {\sqrt[{i}]{i}}=i^{-i}=i^{\frac {1}{i}}=(i^{i})^{-1}=e^{\frac {\pi }{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/904fff5ea95018fde18c45c94097a379edad291e)
![{\displaystyle {\sqrt[{12}]{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bc835f27425fb3140e1f75a5faa35b1e8b9efc35)
![{\displaystyle {\sqrt[{3}]{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9ca071ab504481c2bb76081aacb03f5519930710)
![{\displaystyle {\sqrt[{3}]{1+{\sqrt[{3}]{1+{\sqrt[{3}]{1+{\sqrt[{3}]{1+\cdots }}}}}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fe5c1cba04372927a214a2ce1b1d6b213bb12ee3)
![{\displaystyle \prod _{n=1}^{\infty }\left[{1+{1 \over n(n+2)}}\right]^{\ln n/\ln 2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cbfef25fcd2817842f1c50956dc798248c418be6)
