Value
Name
Graphics
Symbol
LaTeX
Formula
Nº
OEIS
Continued fraction
Year
Web format
0,70444 22009 99165 59273
Carefree constant 2 [ 1]
C
2
{\displaystyle {\mathcal {C))_{2))
∏
n
=
1
∞
(
1
−
1
p
n
(
p
n
+
1
)
)
p
n
:
p
r
i
m
e
{\displaystyle {\underset {p_{n}:\,{prime)){\prod _{n=1}^{\infty }\left(1-{\frac {1}{p_{n}(p_{n}+1)))\right)))}
N[prod[n=1 to ∞] {1 - 1/(prime(n)* (prime(n)+1))}]
OEIS : A065463
[0;1,2,2,1,1,1,1,4,2,1,1,3,703,2,1,1,1,3,5,1,...]
0.70444220099916559273660335032663721
1.84775 90650 22573 51225 [ Mw 1]
Connective constant [ 2] [ 3]
μ
{\displaystyle {\mu ))
2
+
2
=
lim
n
→
∞
c
n
1
/
n
{\displaystyle {\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
{\displaystyle :\;x^{4}-4x^{2}+2=0}
sqrt(2+sqrt(2))
A
OEIS : A179260
[1;1,5,1,1,3,6,1,3,3,10,10,1,1,1,5,2,3,1,1,3,...]
1.84775906502257351225636637879357657
0.30366 30028 98732 65859 [ Mw 2]
Gauss-Kuzmin-Wirsing constant [ 4]
λ
2
{\displaystyle {\lambda }_{2))
lim
n
→
∞
F
n
(
x
)
−
ln
(
1
−
x
)
(
−
λ
)
n
=
Ψ
(
x
)
,
{\displaystyle \lim _{n\to \infty }{\frac {F_{n}(x)-\ln(1-x)}{(-\lambda )^{n))}=\Psi (x),}
where
Ψ
(
x
)
{\displaystyle \Psi (x)}
is an analytic function with
Ψ
(
0
)
=
Ψ
(
1
)
=
0
{\displaystyle \Psi (0)\!=\!\Psi (1)\!=\!0}
.
OEIS : A038517
[0;3,3,2,2,3,13,1,174,1,1,1,2,2,2,1,1,1,2,2,1,...]
1973
0.30366300289873265859744812190155623
1,57079 63267 94896 61923 [ Mw 3]
Favard constant K1 Wallis product [ 5]
π
2
{\displaystyle {\frac {\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 }
Prod[n=1 to ∞] {(4n^2)/(4n^2-1)}
T
OEIS : A069196
[1;1,1,3,31,1,145,1,4,2,8,1,6,1,2,3,1,4,1,5,1...]
1655
1.57079632679489661923132169163975144
1,60669 51524 15291 76378 [ Mw 4]
Erdős–Borwein constant [ 6] [ 7]
E
B
{\displaystyle {E}_{\,B))
∑
m
=
1
∞
∑
n
=
1
∞
1
2
m
n
=
∑
n
=
1
∞
1
2
n
−
1
=
1
1
+
1
3
+
1
7
+
1
15
+
.
.
.
{\displaystyle \sum _{m=1}^{\infty }\sum _{n=1}^{\infty }{\frac {1}{2^{mn))}=\sum _{n=1}^{\infty }{\frac {1}{2^{n}-1))={\frac {1}{1))\!+\!{\frac {1}{3))\!+\!{\frac {1}{7))\!+\!{\frac {1}{15))\!+\!...}
sum[n=1 to ∞] {1/(2^n-1)}
I
OEIS : A065442
[1;1,1,1,1,5,2,1,2,29,4,1,2,2,2,2,6,1,7,1,...]
1949
1.60669515241529176378330152319092458
1.61803 39887 49894 84820 [ Mw 5]
Phi, Golden ratio [ 8]
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
A
OEIS : A001622
[0;1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,...] = [0;1 ,...]
-300 ~
1.61803398874989484820458633436563812
1.64493 40668 48226 43647 [ Mw 6]
Riemann Function 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
OEIS : A013661
[1;1,1,1,4,2,4,7,1,4,2,3,4,10 1,2,1,1,1,15,...]
1826 to 1866
1.64493406684822643647241516664602519
1.73205 08075 68877 29352 [ Mw 7]
Theodorus constant [ 9]
3
{\displaystyle {\sqrt {3))}
3
3
3
3
3
⋯
3
3
3
3
3
{\displaystyle {\sqrt[{3}]{3\,{\sqrt[{3}]{3\,{\sqrt[{3}]{3\,{\sqrt[{3}]{3\,{\sqrt[{3}]{3\,\cdots ))))))))))}
(3(3(3(3(3(3(3) ^1/3)^1/3)^1/3) ^1/3)^1/3)^1/3) ^1/3 ...
A
OEIS : A002194
[1;1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,...] = [1;1,2 ,...]
-465 to -398
1.73205080756887729352744634150587237
1.75793 27566 18004 53270 [ Mw 8]
Kasner number
R
{\displaystyle {R))
1
+
2
+
3
+
4
+
⋯
{\displaystyle {\sqrt {1+{\sqrt {2+{\sqrt {3+{\sqrt {4+\cdots ))))))))}
Fold[Sqrt[#1+#2] &,0,Reverse [Range[20]]]
OEIS : A072449
[1;1,3,7,1,1,1,2,3,1,4,1,1,2,1,2,20,1,2,2,...]
1878 a 1955
1.75793275661800453270881963821813852
2.29558 71493 92638 07403 [ Mw 9]
Universal parabolic constant [ 10]
P
2
{\displaystyle {P}_{\,2))
ln
(
1
+
2
)
+
2
=
arcsinh
(
1
)
+
2
{\displaystyle \ln(1+{\sqrt {2)))+{\sqrt {2))\;=\;\operatorname {arcsinh} (1)+{\sqrt {2))}
ln(1+sqrt 2)+sqrt 2
T
OEIS : A103710
[2;3,2,1,1,1,1,3,3,1,1,4,2,3,2,7,1,6,1,8,7,2,1,...]
2.29558714939263807403429804918949038
1.78657 64593 65922 46345 [ Mw 10]
Silverman constant[ 11]
S
m
{\displaystyle ((\mathcal {S))_{_{m))))
∑
n
=
1
∞
1
ϕ
(
n
)
σ
1
(
n
)
=
∏
n
=
1
∞
(
1
+
∑
k
=
1
∞
1
p
n
2
k
−
p
n
k
−
1
)
p
n
:
p
r
i
m
e
{\displaystyle \sum _{n=1}^{\infty }{\frac {1}{\phi (n)\sigma _{1}(n)))={\underset {p_{n}:\,{prime)){\prod _{n=1}^{\infty }\left(1+\sum _{k=1}^{\infty }{\frac {1}{p_{n}^{2k}-p_{n}^{k-1))}\right)))}
ø() = Euler's totient function , σ1 () = Divisor function .
Sum[n=1 to ∞] {1/[EulerPhi(n) DivisorSigma (1,n)]}
OEIS : A093827
[1;1,3,1,2,5,1,65,11,2,1,2,13,1,4,1,1,1,2,5,4,...]
1.78657645936592246345859047554131575
2.59807 62113 53315 94029 [ Mw 11]
Area of the regular hexagon with side equal to 1 [ 12]
A
6
{\displaystyle {\mathcal {A))_{6))
3
3
2
l
2
{\displaystyle {\frac {3{\sqrt {3))}{2))\,l^{2))
3 sqrt(3)/2
A
OEIS : A104956
[2;1,1,2,20,2,1,1,4,1,1,2,20,2,1,1,4,1,1,2,20,...] [2;1,1,2,20,2,1,1,4 ]
2.59807621135331594029116951225880855
0.66131 70494 69622 33528 [ Mw 12]
Feller-Tornier constant [ 13]
C
F
T
{\displaystyle ((\mathcal {C))_{_{FT))))
1
2
∏
n
=
1
∞
(
1
−
2
p
n
2
)
+
1
2
p
n
:
p
r
i
m
e
=
3
π
2
∏
n
=
1
∞
(
1
−
1
p
n
2
−
1
)
+
1
2
{\displaystyle {\underset {p_{n}:\,{prime))((\frac {1}{2))\prod _{n=1}^{\infty }\left(1-{\frac {2}{p_{n}^{2))}\right){+}{\frac {1}{2))))={\frac {3}{\pi ^{2))}\prod _{n=1}^{\infty }\left(1-{\frac {1}{p_{n}^{2}-1))\right){+}{\frac {1}{2))}
[prod[n=1 to ∞] {1-2/prime(n)^2}] /2 + 1/2
T ?
OEIS : A065493
[0;1,1,1,20,9,1,2,5,1,2,3,2,3,38,8,1,16,2,2,...]
1932
0.66131704946962233528976584627411853
1.46099 84862 06318 35815 [ Mw 13]
Baxter's Four-coloring constant [ 14]
Mapamundi Four-Coloring
C
2
{\displaystyle {\mathcal {C))^{2))
∏
n
=
1
∞
(
3
n
−
1
)
2
(
3
n
−
2
)
(
3
n
)
=
3
4
π
2
Γ
(
1
3
)
3
{\displaystyle \prod _{n=1}^{\infty }{\frac {(3n-1)^{2)){(3n-2)(3n)))={\frac {3}{4\pi ^{2))}\,\Gamma \left({\frac {1}{3))\right)^{3))
Γ() = Gamma function
3×Gamma(1/3) ^3/(4 pi^2)
OEIS : A224273
[1;2,5,1,10,8,1,12,3,1,5,3,5,8,2,1,23,1,2,161,...]
1970
1.46099848620631835815887311784605969
1.92756 19754 82925 30426 [ Mw 14]
Tetranacci constant
T
{\displaystyle {\mathcal {T))}
Positive root of
:
x
4
−
x
3
−
x
2
−
x
−
1
=
0
{\displaystyle :\;\;x^{4}-x^{3}-x^{2}-x-1=0}
Root[x+x^-4-2=0]
OEIS : A086088
[1;1,12,1,4,7,1,21,1,2,1,4,6,1,10,1,2,2,1,7,1,...]
1.92756197548292530426190586173662216
1.00743 47568 84279 37609 [ Mw 15]
DeVicci's tesseract constant
f
(
3
,
4
)
{\displaystyle {f_{(3,4)))}
The largest cube that can pass through in an 4D hypercube.
Positive root of
:
4
x
4
−
28
x
3
−
7
x
2
+
16
x
+
16
=
0
{\displaystyle :\;\;4x^{4}{-}28x^{3}{-}7x^{2}{+}16x{+}16=0}
Root[4*x^8-28*x^6 -7*x^4+16*x^2+16 =0]
A
OEIS : A243309
[1;134,1,1,73,3,1,5,2,1,6,3,11,4,1,5,5,1,1,48,...]
1.00743475688427937609825359523109914
1.70521 11401 05367 76428 [ Mw 16]
Niven's constant [ 15]
C
{\displaystyle {C))
1
+
∑
n
=
2
∞
(
1
−
1
ζ
(
n
)
)
{\displaystyle 1+\sum _{n=2}^{\infty }\left(1-{\frac {1}{\zeta (n)))\right)}
1+ Sum[n=2 to ∞] {1-(1/Zeta(n))}
OEIS : A033150
[1;1,2,2,1,1,4,1,1,3,4,4,8,4,1,1,2,1,1,11,1,...]
1969
1.70521114010536776428855145343450816
0.60459 97880 78072 61686 [ Mw 17]
Relationship among the area of an equilateral triangle and the inscribed circle.
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 }
Dirichlet series
Sum[1/(n Binomial[2 n, n]) , {n, 1, ∞}]
T
OEIS : A073010
[0;1,1,1,1,8,10,2,2,3,3,1,9,2,5,4,1,27,27,6,6,...]
0.60459978807807261686469275254738524
1.15470 05383 79251 52901 [ Mw 18]
Hermite Constant [ 16]
γ
2
{\displaystyle \gamma _{_{2))}
2
3
=
1
cos
(
π
6
)
{\displaystyle {\frac {2}{\sqrt {3))}={\frac {1}{\cos \,({\frac {\pi }{6)))))}
2/sqrt(3)
A
1+OEIS : A246724
[1;6,2,6,2,6,2,6,2,6,2,6,2,6,2,6,2,6,2,6,2,6,2,...] [1;6,2 ]
1.15470053837925152901829756100391491
0.41245 40336 40107 59778 [ Mw 19]
Prouhet–Thue–Morse constant [ 17]
τ
{\displaystyle \tau }
∑
n
=
0
∞
t
n
2
n
+
1
{\displaystyle \sum _{n=0}^{\infty }{\frac {t_{n)){2^{n+1))))
where
t
n
{\displaystyle {t_{n))}
is the Thue–Morse sequence and Where
τ
(
x
)
=
∑
n
=
0
∞
(
−
1
)
t
n
x
n
=
∏
n
=
0
∞
(
1
−
x
2
n
)
{\displaystyle \tau (x)=\sum _{n=0}^{\infty }(-1)^{t_{n))\,x^{n}=\prod _{n=0}^{\infty }(1-x^{2^{n)))}
T
OEIS : A014571
[0;2,2,2,1,4,3,5,2,1,4,2,1,5,44,1,4,1,2,4,1,1,...]
0.41245403364010759778336136825845528
0.58057 75582 04892 40229 [ Mw 20]
Pell Constant [ 18]
P
P
e
l
l
{\displaystyle ((\mathcal {P))_{_{Pell))))
1
−
∏
n
=
0
∞
(
1
−
1
2
2
n
+
1
)
{\displaystyle 1-\prod _{n=0}^{\infty }\left(1-{\frac {1}{2^{2n+1))}\right)}
N[1-prod[n=0 to ∞] {1-1/(2^(2n+1)}]
T ?
OEIS : A141848
[0;1,1,2,1,1,1,1,14,1,3,1,1,6,9,18,7,1,27,1,1,...]
0.58057755820489240229004389229702574
0.66274 34193 49181 58097 [ Mw 21]
Laplace limit [ 19]
λ
{\displaystyle {\lambda ))
x
e
x
2
+
1
x
2
+
1
+
1
=
1
{\displaystyle {\frac {x\;e^{\sqrt {x^{2}+1))}((\sqrt {x^{2}+1))+1))=1}
(x e^sqrt(x^2+1)) /(sqrt(x^2+1)+1) = 1
OEIS : A033259
[0;1,1,1,27,1,1,1,8,2,154,2,4,1,5,1,1,2,1601,...]
1782 ~
0.66274341934918158097474209710925290
0.17150 04931 41536 06586 [ Mw 22]
Hall-Montgomery Constant [ 20]
δ
0
{\displaystyle ((\delta }_{_{0))))
1
+
π
2
6
+
2
L
i
2
(
−
e
)
L
i
2
= Dilogarithm integral
{\displaystyle 1+{\frac {\pi ^{2)){6))+2\;\mathrm {Li} _{2}\left(-{\sqrt {e))\;\right)\quad \mathrm {Li} _{2}\,\scriptstyle {\text{= Dilogarithm integral))}
1 + Pi^2/6 + 2*PolyLog[2, -Sqrt[E]]
OEIS : A143301
[0;5,1,4,1,10,1,1,11,18,1,2,19,14,1,51,1,2,1,...]
0.17150049314153606586043997155521210
1.55138 75245 48320 39226 [ Mw 23]
Calabi triangle constant [ 21]
C
C
R
{\displaystyle {C_{_{CR))))
1
3
+
(
−
23
+
3
i
237
)
1
3
3
⋅
2
2
3
+
11
3
(
2
(
−
23
+
3
i
237
)
)
1
3
{\displaystyle {1 \over 3}+{(-23+3i{\sqrt {237)))^{\tfrac {1}{3)) \over 3\cdot 2^{\tfrac {2}{3))}+{11 \over 3(2(-23+3i{\sqrt {237))))^{\tfrac {1}{3))))
FindRoot[ 2x^3-2x^2-3x+2 ==0, {x, 1.5}, WorkingPrecision->40]
A
OEIS : A046095
[1;1,1,4,2,1,2,1,5,2,1,3,1,1,390,1,1,2,11,6,2,...]
1946 ~
1.55138752454832039226195251026462381
1.22541 67024 65177 64512 [ Mw 24]
Gamma(3/4) [ 22]
Γ
(
3
4
)
{\displaystyle \Gamma ({\tfrac {3}{4)))}
(
−
1
+
3
4
)
!
=
(
−
1
4
)
!
{\displaystyle \left(-1+{\frac {3}{4))\right)!=\left(-{\frac {1}{4))\right)!}
(-1+3/4)!
OEIS : A068465
[1;4,2,3,2,2,1,1,1,2,1,4,7,1,171,3,2,3,1,1,8,3,...]
1.22541670246517764512909830336289053
1.20205 69031 59594 28539 [ Mw 25]
Apéry's constant [ 23]
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 =}
1
2
∑
n
=
1
∞
H
n
n
2
=
1
2
∑
i
=
1
∞
∑
j
=
1
∞
1
i
j
(
i
+
j
)
=
∫
0
1
∫
0
1
∫
0
1
d
x
d
y
d
z
1
−
x
y
z
{\displaystyle {\frac {1}{2))\sum _{n=1}^{\infty }{\frac {H_{n)){n^{2))}={\frac {1}{2))\sum _{i=1}^{\infty }\sum _{j=1}^{\infty }{\frac {1}{ij(i{+}j)))=\!\!\int \limits _{0}^{1}\!\!\int \limits _{0}^{1}\!\!\int \limits _{0}^{1}{\frac {\mathrm {d} x\mathrm {d} y\mathrm {d} z}{1-xyz))}
Sum[n=1 to ∞] {1/n^3}
I
OEIS : A010774
[1;4,1,18,1,1,1,4,1,9,9,2,1,1,1,2,7,1,1,7,11,...]
1979
1.20205690315959428539973816151144999
0.91596 55941 77219 01505 [ Mw 26]
Catalan's constant [ 24] [ 25] [ 26]
C
{\displaystyle {C))
∫
0
1
∫
0
1
1
1
+
x
2
y
2
d
x
d
y
=
∑
n
=
0
∞
(
−
1
)
n
(
2
n
+
1
)
2
=
1
1
2
−
1
3
2
+
⋯
{\displaystyle \int _{0}^{1}\!\!\int _{0}^{1}\!\!{\frac {1}{1{+}x^{2}y^{2))}\,dx\,dy=\!\sum _{n=0}^{\infty }\!{\frac {(-1)^{n)){(2n{+}1)^{2))}\!=\!{\frac {1}{1^{2))}{-}{\frac {1}{3^{2))}{+}{\cdots ))
Sum[n=0 to ∞] {(-1)^n/(2n+1)^2}
T
OEIS : A006752
[0;1,10,1,8,1,88,4,1,1,7,22,1,2,3,26,1,11,...]
1864
0.91596559417721901505460351493238411
0.78539 81633 97448 30961 [ Mw 27]
Beta(1) [ 27]
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
OEIS : A003881
[0; 1,3,1,1,1,15,2,72,1,9,1,17,1,2,1,5,1,1,10,...]
1805 to 1859
0.78539816339744830961566084581987572
0.00131 76411 54853 17810 [ Mw 28]
Heath-Brown–Moroz constant [ 28]
C
H
B
M
{\displaystyle {C_{_{HBM))))
∏
n
=
1
∞
(
1
−
1
p
n
)
7
(
1
+
7
p
n
+
1
p
n
2
)
p
n
:
p
r
i
m
e
{\displaystyle {\underset {p_{n}:\,{prime)){\prod _{n=1}^{\infty }\left(1-{\frac {1}{p_{n))}\right)^{7}\left(1+{\frac {7p_{n}+1}{p_{n}^{2))}\right)))}
N[prod[n=1 to ∞] {((1-1/prime(n))^7) *(1+(7*prime(n)+1) /(prime(n)^2))}]
T ?
OEIS : A118228
[0,0,1,3,1,7,6,4,1,1,5,4,8,5,3,1,7,8,1,0,9,8,1,...]
0.00131764115485317810981735232251358
0.56755 51633 06957 82538
Module of Infinite 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^...)
OEIS : A212479
[0;1,1,3,4,1,58,12,1,51,1,4,12,1,1,2,2,3,...]
0.56755516330695782538461314419245334
0.78343 05107 12134 40705 [ Mw 29]
Sophomore's dream 1 J.Bernoulli [ 29]
I
1
{\displaystyle {I}_{1))
∫
0
1
x
x
d
x
=
∑
n
=
1
∞
(
−
1
)
n
+
1
n
n
=
1
1
1
−
1
2
2
+
1
3
3
−
⋯
{\displaystyle \int _{0}^{1}\!x^{x}\,dx=\sum _{n=1}^{\infty }{\frac {(-1)^{n+1)){n^{n))}={\frac {1}{1^{1))}-{\frac {1}{2^{2))}+{\frac {1}{3^{3))}-{\cdots ))
Sum[n=1 to ∞] {-(-1)^n /n^n}
OEIS : A083648
[0;1,3,1,1,1,1,1,1,2,4,7,2,1,2,1,1,1,2,1,14,...]
1697
0.78343051071213440705926438652697546
1.29128 59970 62663 54040 [ Mw 30]
Sophomore's dream 2 J.Bernoulli [ 30]
I
2
{\displaystyle {I}_{2))
∫
0
1
1
x
x
d
x
=
∑
n
=
1
∞
1
n
n
=
1
1
1
+
1
2
2
+
1
3
3
+
1
4
4
+
⋯
{\displaystyle \int _{0}^{1}\!{\frac {1}{x^{x))}\,dx=\sum _{n=1}^{\infty }{\frac {1}{n^{n))}={\frac {1}{1^{1))}+{\frac {1}{2^{2))}+{\frac {1}{3^{3))}+{\frac {1}{4^{4))}+\cdots }
Sum[n=1 to ∞] {1/(n^n)}
OEIS : A073009
[1;3,2,3,4,3,1,2,1,1,6,7,2,5,3,1,2,1,8,1,2,4,...]
1697
1.29128599706266354040728259059560054
0.70523 01717 91800 96514 [ Mw 31]
Primorial constant Sum of the product of inverse of primes [ 31]
P
#
{\displaystyle {P_{\#))}
∑
n
=
1
∞
1
p
n
#
=
1
2
+
1
6
+
1
30
+
1
210
+
.
.
.
=
∑
k
=
1
∞
∏
n
=
1
k
1
p
n
p
n
:
p
r
i
m
e
{\displaystyle {\underset {p_{n}:\,{prime)){\sum _{n=1}^{\infty }{\frac {1}{p_{n}\#))={\frac {1}{2))+{\frac {1}{6))+{\frac {1}{30))+{\frac {1}{210))+...=\sum _{k=1}^{\infty }\prod _{n=1}^{k}{\frac {1}{p_{n))))))
Sum[k=1 to ∞] (prod[n=1 to k] {1/prime(n)})
OEIS : A064648
[0;1,2,2,1,1,4,1,2,1,1,6,13,1,4,1,16,6,1,1,4,...]
0.70523017179180096514743168288824851
0.14758 36176 50433 27417 [ Mw 32]
Plouffe's gamma constant [ 32]
C
{\displaystyle {C))
1
π
arctan
1
2
=
1
π
∑
n
=
0
∞
(
−
1
)
n
(
2
2
n
+
1
)
(
2
n
+
1
)
{\displaystyle {\frac {1}{\pi ))\arctan {\frac {1}{2))={\frac {1}{\pi ))\sum _{n=0}^{\infty }{\frac {(-1)^{n)){(2^{2n+1})(2n+1)))}
=
1
π
(
1
2
−
1
3
⋅
2
3
+
1
5
⋅
2
5
−
1
7
⋅
2
7
+
⋯
)
{\displaystyle ={\frac {1}{\pi ))\left({\frac {1}{2))-{\frac {1}{3\cdot 2^{3))}+{\frac {1}{5\cdot 2^{5))}-{\frac {1}{7\cdot 2^{7))}+\cdots \right)}
Arctan(1/2)/pi
T
OEIS : A086203
[0;6,1,3,2,5,1,6,5,3,1,1,2,1,1,2,3,1,2,3,2,2,...]
0.14758361765043327417540107622474052
0.15915 49430 91895 33576 [ Mw 33]
Plouffe's A constant [ 33]
A
{\displaystyle {A))
1
2
π
{\displaystyle {\frac {1}{2\pi ))}
1/(2 pi)
T
OEIS : A086201
[0;6,3,1,1,7,2,146,3,6,1,1,2,7,5,5,1,4,1,2,42,...]
0.15915494309189533576888376337251436
0.29156 09040 30818 78013 [ Mw 34]
Dimer constant 2D, Domino tiling [ 34] [ 35]
C
π
{\displaystyle {\frac {C}{\pi ))}
C=Catalan
∫
−
π
π
cosh
−
1
(
cos
(
t
)
+
3
2
)
4
π
d
t
{\displaystyle \int \limits _{-\pi }^{\pi }{\frac {\cosh ^{-1}\left({\frac {\sqrt {\cos(t)+3)){\sqrt {2))}\right)}{4\pi ))\,dt}
N[int[-pi to pi] {arccosh(sqrt( cos(t)+3)/sqrt(2)) /(4*Pi)dt}]
OEIS : A143233
[0;3,2,3,16,8,10,3,1,1,2,1,3,1,2,13,1,1,4,1,5,...]
0.29156090403081878013838445646839491
0.49801 56681 18356 04271
0.15494 98283 01810 68512 i
Factorial (i )[ 36]
i
!
{\displaystyle {i}\,!}
Γ
(
1
+
i
)
=
i
Γ
(
i
)
=
∫
0
∞
t
i
e
t
d
t
{\displaystyle \Gamma (1+i)=i\,\Gamma (i)=\int \limits _{0}^{\infty }{\frac {t^{i)){e^{t))}\mathrm {d} t}
Integral_0^∞ t^i/e^t dt
C
OEIS : A212877 OEIS : A212878
[0;6,2,4,1,8,1,46,2,2,3,5,1,10,7,5,1,7,2,...] - [0;2,125,2,18,1,2,1,1,19,1,1,1,2,3,34,...] i
0.49801566811835604271369111746219809 - 0.15494982830181068512495513048388 i
2.09455 14815 42326 59148 [ Mw 35]
Wallis Constant
W
{\displaystyle W}
45
−
1929
18
3
+
45
+
1929
18
3
{\displaystyle {\sqrt[{3}]{\frac {45-{\sqrt {1929))}{18))}+{\sqrt[{3}]{\frac {45+{\sqrt {1929))}{18))))
(((45-sqrt(1929)) /18))^(1/3)+ (((45+sqrt(1929)) /18))^(1/3)
T
OEIS : A007493
[2;10,1,1,2,1,3,1,1,12,3,5,1,1,2,1,6,1,11,4,...]
1616 to 1703
2.09455148154232659148238654057930296
0.72364 84022 98200 00940 [ Mw 36]
Sarnak constant
C
s
a
{\displaystyle {C_{sa))}
∏
p
>
2
(
1
−
p
+
2
p
3
)
{\displaystyle \prod _{p>2}{\Big (}1-{\frac {p+2}{p^{3))}{\Big )))
N[prod[k=2 to ∞] {1-(prime(k)+2) /(prime(k)^3)}]
T ?
OEIS : A065476
[0;1,2,1,1,1,1,1,1,1,4,4,1,1,1,1,1,1,1,8,2,1,1,...]
0.72364840229820000940884914980912759
0.63212 05588 28557 67840 [ Mw 37]
Time constant [ 37]
τ
{\displaystyle {\tau ))
lim
n
→
∞
1
−
!
n
n
!
=
lim
n
→
∞
P
(
n
)
=
∫
0
1
e
−
x
d
x
=
1
−
1
e
=
{\displaystyle \lim _{n\to \infty }1-{\frac {!n}{n!))=\lim _{n\to \infty }P(n)=\int _{0}^{1}e^{-x}dx=1{-}{\frac {1}{e))=}
∑
n
=
1
∞
(
−
1
)
n
+
1
n
!
=
1
1
!
−
1
2
!
+
1
3
!
−
1
4
!
+
1
5
!
−
1
6
!
+
⋯
{\displaystyle \sum \limits _{n=1}^{\infty }{\frac {(-1)^{n+1)){n!))={\frac {1}{1!)){-}{\frac {1}{2!)){+}{\frac {1}{3!)){-}{\frac {1}{4!)){+}{\frac {1}{5!)){-}{\frac {1}{6!)){+}\cdots }
lim_(n->∞) (1- !n/n!) !n=subfactorial
T
OEIS : A068996
[0;1,1,1,2,1,1,4,1,1,6,1,1,8,1,1,10,1,1,12,1,...] = [0;1,1,1,2n ], n∈ℕ
0.63212055882855767840447622983853913
1.04633 50667 70503 18098
Minkowski-Siegel mass constant [ 38]
F
1
{\displaystyle F_{1))
∏
n
=
1
∞
n
!
2
π
n
(
n
e
)
n
1
+
1
n
12
{\displaystyle \prod _{n=1}^{\infty }{\frac {n!}((\sqrt {2\pi n))\left({\frac {n}{e))\right)^{n}{\sqrt[{12}]{1+{\tfrac {1}{n))))))}
N[prod[n=1 to ∞] n! /(sqrt(2*Pi*n) *(n/e)^n *(1+1/n) ^(1/12))]
OEIS : A213080
[1;21,1,1,2,1,1,4,2,1,5,7,2,1,20,1,1,1134,3,..]
1867 1885 1935
1.04633506677050318098095065697776037
5.24411 51085 84239 62092 [ Mw 38]
Lemniscate Constant [ 39]
2
ϖ
{\displaystyle 2\varpi }
[
Γ
(
1
4
)
]
2
2
π
=
4
∫
0
1
d
x
(
1
−
x
2
)
(
2
−
x
2
)
{\displaystyle {\frac {[\Gamma ({\tfrac {1}{4)))]^{2)){\sqrt {2\pi ))}=4\int _{0}^{1}{\frac {dx}{\sqrt {(1-x^{2})(2-x^{2})))))
Gamma[ 1/4 ]^2 /Sqrt[ 2 Pi ]
OEIS : A064853
[5;4,10,2,1,2,3,29,4,1,2,1,2,1,2,1,4,9,1,4,1,2,...]
1718
5.24411510858423962092967917978223883
0.66170 71822 67176 23515 [ Mw 39]
Robbins constant [ 40]
Δ
(
3
)
{\displaystyle \Delta (3)}
4
+
17
2
−
6
3
−
7
π
105
+
ln
(
1
+
2
)
5
+
2
ln
(
2
+
3
)
5
{\displaystyle {\frac {4\!+\!17{\sqrt {2))\!-6{\sqrt {3))\!-7\pi }{105))\!+\!{\frac {\ln(1\!+\!{\sqrt {2)))}{5))\!+\!{\frac {2\ln(2\!+\!{\sqrt {3)))}{5))}
(4+17*2^(1/2)-6 *3^(1/2)+21*ln(1+ 2^(1/2))+42*ln(2+ 3^(1/2))-7*Pi)/105
OEIS : A073012
[0;1,1,1,21,1,2,1,4,10,1,2,2,1,3,11,1,331,1,4,...]
1978
0.66170718226717623515583113324841358
1.30357 72690 34296 39125 [ Mw 40]
Conway constant [ 41]
λ
{\displaystyle {\lambda ))
x
71
−
x
69
−
2
x
68
−
x
67
+
2
x
66
+
2
x
65
+
x
64
−
x
63
−
x
62
−
x
61
−
x
60
−
x
59
+
2
x
58
+
5
x
57
+
3
x
56
−
2
x
55
−
10
x
54
−
3
x
53
−
2
x
52
+
6
x
51
+
6
x
50
+
x
49
+
9
x
48
−
3
x
47
−
7
x
46
−
8
x
45
−
8
x
44
+
10
x
43
+
6
x
42
+
8
x
41
−
5
x
40
−
12
x
39
+
7
x
38
−
7
x
37
+
7
x
36
+
x
35
−
3
x
34
+
10
x
33
+
x
32
−
6
x
31
−
2
x
30
−
10
x
29
−
3
x
28
+
2
x
27
+
9
x
26
−
3
x
25
+
14
x
24
−
8
x
23
−
7
x
21
+
9
x
20
+
3
x
19
−
4
x
18
−
10
x
17
−
7
x
16
+
12
x
15
+
7
x
14
+
2
x
13
−
12
x
12
−
4
x
11
−
2
x
10
+
5
x
9
+
x
7
−
7
x
6
+
7
x
5
−
4
x
4
+
12
x
3
−
6
x
2
+
3
x
−
6
=
0
{\displaystyle {\begin{smallmatrix}x^{71}\quad \ -x^{69}-2x^{68}-x^{67}+2x^{66}+2x^{65}+x^{64}-x^{63}-x^{62}-x^{61}-x^{60}\\-x^{59}+2x^{58}+5x^{57}+3x^{56}-2x^{55}-10x^{54}-3x^{53}-2x^{52}+6x^{51}+6x^{50}\\+x^{49}+9x^{48}-3x^{47}-7x^{46}-8x^{45}-8x^{44}+10x^{43}+6x^{42}+8x^{41}-5x^{40}\\-12x^{39}+7x^{38}-7x^{37}+7x^{36}+x^{35}-3x^{34}+10x^{33}+x^{32}-6x^{31}-2x^{30}\\-10x^{29}-3x^{28}+2x^{27}+9x^{26}-3x^{25}+14x^{24}-8x^{23}\quad \ -7x^{21}+9x^{20}\\+3x^{19}\!-4x^{18}\!-10x^{17}\!-7x^{16}\!+12x^{15}\!+7x^{14}\!+2x^{13}\!-12x^{12}\!-4x^{11}\!-2x^{10}\\+5x^{9}+x^{7}\quad \ -7x^{6}+7x^{5}-4x^{4}+12x^{3}-6x^{2}+3x-6\ =\ 0\quad \quad \quad \end{smallmatrix))}
A
OEIS : A014715
[1;3,3,2,2,54,5,2,1,16,1,30,1,1,1,2,2,1,14,1,...]
1987
1.30357726903429639125709911215255189
1.18656 91104 15625 45282 [ Mw 41]
Khinchin–Lévy constant [ 42]
β
{\displaystyle {\beta ))
π
2
12
ln
2
{\displaystyle {\frac {\pi ^{2)){12\,\ln 2))}
pi^2 /(12 ln 2)
OEIS : A100199
[1;5,2,1,3,1,1,28,18,16,3,2,6,2,6,1,1,5,5,9,...]
1935
1.18656911041562545282172297594723712
0.83564 88482 64721 05333
Baker constant [ 43]
β
3
{\displaystyle \beta _{3))
∫
0
1
d
t
1
+
t
3
=
∑
n
=
0
∞
(
−
1
)
n
3
n
+
1
=
1
3
(
ln
2
+
π
3
)
{\displaystyle \int _{0}^{1}{\frac {\mathrm {d} t}{1+t^{3))}=\sum _{n=0}^{\infty }{\frac {(-1)^{n)){3n+1))={\frac {1}{3))\left(\ln 2+{\frac {\pi }{\sqrt {3))}\right)}
Sum[n=0 to ∞] {((-1)^(n))/(3n+1)}
OEIS : A113476
[0;1,5,11,1,4,1,6,1,4,1,1,1,2,1,3,2,2,2,2,1,3,...]
0.83564884826472105333710345970011076
23.10344 79094 20541 6160 [ Mw 42]
Kempner Serie (0) [ 44]
K
0
{\displaystyle {K_{0))}
1
+
1
2
+
1
3
+
⋯
+
1
9
+
1
11
+
⋯
+
1
19
+
1
21
+
⋯
+
etc.
{\displaystyle 1{+}{\frac {1}{2)){+}{\frac {1}{3)){+}\cdots {+}{\frac {1}{9)){+}{\frac {1}{11)){+}\cdots {+}{\frac {1}{19)){+}{\frac {1}{21)){+}\cdots {+}\,{\text{etc.))}
+
1
99
+
1
111
+
⋯
+
1
119
+
1
121
+
⋯
d
e
n
o
m
i
n
a
t
o
r
s
c
o
n
t
a
i
n
i
n
g
0.
E
x
c
l
u
d
i
n
g
a
l
l
{\displaystyle {+}{\frac {1}{99)){+}{\frac {1}{111)){+}\cdots {+}{\frac {1}{119)){+}{\frac {1}{121)){+}\cdots \;\;{\overset {Excluding\;all}{\underset {containing\;0.}{\scriptstyle denominators))))
1+1/2+1/3+1/4+1/5 +1/6+1/7+1/8+1/9 +1/11+1/12+1/13 +1/14+1/15+...
OEIS : A082839
[23;9,1,2,3244,1,1,5,1,2,2,8,3,1,1,6,1,84,1,...]
23.1034479094205416160340540433255981
0.98943 12738 31146 95174 [ Mw 43]
Lebesgue constant [ 45]
C
1
{\displaystyle {C_{1))}
lim
n
→
∞
(
L
n
−
4
π
2
ln
(
2
n
+
1
)
)
=
4
π
2
(
∑
k
=
1
∞
2
ln
k
4
k
2
−
1
−
Γ
′
(
1
2
)
Γ
(
1
2
)
)
{\displaystyle \lim _{n\to \infty }\!\!\left(\!{L_{n}{-}{\frac {4}{\pi ^{2))}\ln(2n{+}1)}\!\!\right)\!{=}{\frac {4}{\pi ^{2))}\!\left({\sum _{k=1}^{\infty }\!{\frac {2\ln k}{4k^{2}{-}1))}{-}{\frac {\Gamma '({\tfrac {1}{2)))}{\Gamma ({\tfrac {1}{2)))))\!\!\right)}
4/pi^2*[(2 Sum[k=1 to ∞] {ln(k)/(4*k^2-1)}) -poligamma(1/2)]
OEIS : A243277
[0;1,93,1,1,1,1,1,1,1,7,1,12,2,15,1,2,7,2,1,5,...]
?
0.98943127383114695174164880901886671
0.19452 80494 65325 11361 [ Mw 44]
2nd du Bois-Reymond constant [ 46]
C
2
{\displaystyle {C_{2))}
e
2
−
7
2
=
∫
0
∞
|
d
d
t
(
sin
t
t
)
n
|
d
t
−
1
{\displaystyle {\frac {e^{2}-7}{2))=\int _{0}^{\infty }\left|((\frac {d}{dt))\left({\frac {\sin t}{t))\right)^{n))\right|\,dt-1}
(e^2-7)/2
T
OEIS : A062546
[0;5,7,9,11,13,15,17,19,21,23,25,27,29,31,...] = [0;2p+3 ], p∈ℕ
0.19452804946532511361521373028750390
0.78853 05659 11508 96106 [ Mw 45]
Lüroth constant[ 47]
C
L
{\displaystyle C_{L))
∑
n
=
2
∞
ln
(
n
n
−
1
)
n
{\displaystyle \sum _{n=2}^{\infty }{\frac {\ln \left({\frac {n}{n-1))\right)}{n))}
Sum[n=2 to ∞] log(n/(n-1))/n
OEIS : A085361
[0;1,3,1,2,1,2,4,1,127,1,2,2,1,3,8,1,1,2,1,16,...]
0.78853056591150896106027632216944432
1.18745 23511 26501 05459 [ Mw 46]
Foias constant α [ 48]
F
α
{\displaystyle F_{\alpha ))
x
n
+
1
=
(
1
+
1
x
n
)
n
for
n
=
1
,
2
,
3
,
…
{\displaystyle x_{n+1}=\left(1+{\frac {1}{x_{n))}\right)^{n}{\text{ for ))n=1,2,3,\ldots }
Foias constant is the unique real number such that if x1 = α then the sequence diverges to ∞. When x 1 = α ,
lim
n
→
∞
x
n
log
n
n
=
1
{\displaystyle \,\lim _{n\to \infty }x_{n}{\tfrac {\log n}{n))=1}
OEIS : A085848
[1;5,2,1,81,3,2,2,1,1,1,1,1,6,1,1,3,1,1,4,3,2,...]
2000
1.18745235112650105459548015839651935
2.29316 62874 11861 03150 [ Mw 47]
Foias constant β
F
β
{\displaystyle F_{\beta ))
x
x
+
1
=
(
x
+
1
)
x
{\displaystyle x^{x+1}=(x+1)^{x))
x^(x+1) = (x+1)^x
OEIS : A085846
[2;3,2,2,3,4,2,3,2,130,1,1,1,1,1,6,3,2,1,15,1,...]
2000
2.29316628741186103150802829125080586
0.82246 70334 24113 21823 [ Mw 48]
Nielsen-Ramanujan constant [ 49]
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))}{-}\cdots }
Sum[n=1 to ∞] {((-1)^(n+1))/n^2}
T
OEIS : A072691
[0;1,4,1,1,1,2,1,1,1,1,3,2,2,4,1,1,1,1,1,1,4...]
1909
0.82246703342411321823620758332301259
0.69314 71805 59945 30941 [ Mw 49]
Natural logarithm of 2 [ 50]
L
n
(
2
)
{\displaystyle Ln(2)}
∑
n
=
1
∞
1
n
2
n
=
∑
n
=
1
∞
(
−
1
)
n
+
1
n
=
1
1
−
1
2
+
1
3
−
1
4
+
⋯
{\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n2^{n))}=\sum _{n=1}^{\infty }{\frac {({-}1)^{n+1)){n))={\frac {1}{1))-{\frac {1}{2))+{\frac {1}{3))-{\frac {1}{4))+{\cdots ))
Sum[n=1 to ∞] {(-1)^(n+1)/n}
T
OEIS : A002162
[0;1,2,3,1,6,3,1,1,2,1,1,1,1,3,10,1,1,1,2,1,1,...]
1550 to 1617
0.69314718055994530941723212145817657
0.47494 93799 87920 65033 [ Mw 50]
Weierstrass constant [ 51]
σ
(
1
2
)
{\displaystyle \sigma ({\tfrac {1}{2)))}
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)
OEIS : A094692
[0;2,9,2,11,1,6,1,4,6,3,19,9,217,1,2,4,8,6...]
1872 ?
0.47494937998792065033250463632798297
0.57721 56649 01532 86060 [ Mw 51]
Euler-Mascheroni constant
γ
{\displaystyle {\gamma ))
∑
n
=
1
∞
∑
k
=
0
∞
(
−
1
)
k
2
n
+
k
=
∑
n
=
1
∞
(
1
n
−
ln
(
1
+
1
n
)
)
{\displaystyle \sum _{n=1}^{\infty }\sum _{k=0}^{\infty }{\frac {(-1)^{k)){2^{n}+k))=\sum _{n=1}^{\infty }\left({\frac {1}{n))-\ln \left(1+{\frac {1}{n))\right)\right)}
=
∫
0
1
−
ln
(
ln
1
x
)
d
x
=
−
Γ
′
(
1
)
=
−
Ψ
(
1
)
{\displaystyle =\int _{0}^{1}-\ln \left(\ln {\frac {1}{x))\right)\,dx=-\Gamma '(1)=-\Psi (1)}
sum[n=1 to ∞] |sum[k=0 to ∞] {((-1)^k)/(2^n+k)}
OEIS : A001620
[0;1,1,2,1,2,1,4,3,13,5,1,1,8,1,2,4,1,1,40,1,11,...]
1735
0.57721566490153286060651209008240243
1.38135 64445 18497 79337
Beta, Kneser-Mahler polynomial constant[ 52]
β
{\displaystyle \beta }
e
2
π
∫
0
π
3
t
tan
t
d
t
=
e
∫
−
1
3
1
3
ln
⌊
1
+
e
2
π
i
t
⌋
d
t
{\displaystyle e^{^{\textstyle {\frac {2}{\pi ))\displaystyle {\int _{0}^{\frac {\pi }{3))}\textstyle {t\tan t\ dt))}=e^{^{\displaystyle {\,\int _{\frac {-1}{3))^{\frac {1}{3))}\textstyle {\,\ln \lfloor 1+e^{2\pi it))\rfloor dt))}
e^((PolyGamma(1,4/3) - PolyGamma(1,2/3) +9)/(4*sqrt(3)*Pi))
OEIS : A242710
[1;2,1,1,1,1,1,4,1,139,2,1,3,5,16,2,1,1,7,2,1,...]
1963
1.38135644451849779337146695685062412
1.35845 62741 82988 43520 [ Mw 52]
Golden Spiral
c
{\displaystyle c}
φ
2
π
=
(
1
+
5
2
)
2
π
{\displaystyle \varphi ^{\frac {2}{\pi ))=\left({\frac {1+{\sqrt {5))}{2))\right)^{\frac {2}{\pi ))}
GoldenRatio^(2/pi)
OEIS : A212224
[1;2,1,3,1,3,10,8,1,1,8,1,15,6,1,3,1,1,2,3,1,1,...]
1.35845627418298843520618060050187945
0.57595 99688 92945 43964 [ Mw 53]
Stephens constant [ 53]
C
S
{\displaystyle C_{S))
∏
n
=
1
∞
(
1
−
p
p
3
−
1
)
{\displaystyle \prod _{n=1}^{\infty }\left(1-{\frac {p}{p^{3}-1))\right)}
Prod[n=1 to ∞] {1-hprime(n) /(hprime(n)^3-1)}
T ?
OEIS : A065478
[0;1,1,2,1,3,1,3,1,2,1,77,2,1,1,10,2,1,1,1,7,...]
?
0.57595996889294543964316337549249669
0.73908 51332 15160 64165 [ Mw 54]
Dottie number [ 54]
d
{\displaystyle d}
lim
x
→
∞
cos
x
(
c
)
=
lim
x
→
∞
cos
(
cos
(
cos
(
⋯
(
cos
(
c
)
)
)
)
)
⏟
x
{\displaystyle \lim _{x\to \infty }\cos ^{x}(c)=\lim _{x\to \infty }\underbrace {\cos(\cos(\cos(\cdots (\cos(c)))))} _{x))
cos(c)=c
OEIS : A003957
[0;1,2,1,4,1,40,1,9,4,2,1,15,2,12,1,21,1,17,...]
?
0.73908513321516064165531208767387340
0.67823 44919 17391 97803 [ Mw 55]
Taniguchi constant [ 55]
C
T
{\displaystyle C_{T))
∏
n
=
1
∞
(
1
−
3
p
n
3
+
2
p
n
4
+
1
p
n
5
−
1
p
n
6
)
{\displaystyle \prod _{n=1}^{\infty }\left(1-{\frac {3}((p_{n))^{3))}+{\frac {2}((p_{n))^{4))}+{\frac {1}((p_{n))^{5))}-{\frac {1}((p_{n))^{6))}\right)}
p
n
=
prime
{\displaystyle \scriptstyle p_{n}=\,{\text{prime))}
Prod[n=1 to ∞] {1 -3/ithprime(n)^3 +2/ithprime(n)^4 +1/ithprime(n)^5 -1/ithprime(n)^6}
T ?
OEIS : A175639
[0;1,2,9,3,1,2,9,11,1,13,2,15,1,1,1,2,4,1,1,1,...]
?
0.67823449191739197803553827948289481
1.85407 46773 01371 91843 [ Mw 56]
Gauss' Lemniscate constant[ 56]
L
/
2
{\displaystyle L{\text{/)){\sqrt {2))}
∫
0
∞
d
x
1
+
x
4
=
1
4
π
Γ
(
1
4
)
2
=
4
(
1
4
!
)
2
π
{\displaystyle \int \limits _{0}^{\infty }{\frac {\mathrm {d} x}{\sqrt {1+x^{4))))={\frac {1}{4{\sqrt {\pi ))))\,\Gamma \left({\frac {1}{4))\right)^{2}={\frac {4\left({\frac {1}{4))!\right)^{2)){\sqrt {\pi ))))
Γ
(
)
= Gamma function
{\displaystyle \scriptstyle \Gamma (){\text{= Gamma function))}
pi^(3/2)/(2 Gamma(3/4)^2)
OEIS : A093341
[1;1,5,1,5,1,3,1,6,2,1,4,16,3,112,2,1,1,18,1,...]
1.85407467730137191843385034719526005
1.75874 36279 51184 82469
Infinite product constant, with Alladi-Grinstead [ 57]
P
r
1
{\displaystyle Pr_{1))
∏
n
=
2
∞
(
1
+
1
n
)
1
n
{\displaystyle \prod _{n=2}^{\infty }{\Big (}1+{\frac {1}{n)){\Big )}^{\frac {1}{n))}
Prod[n=2 to inf] {(1+1/n)^(1/n)}
OEIS : A242623
[1;1,3,6,1,8,1,4,3,1,4,1,1,1,6,5,2,40,1,387,2,...]
1977
1.75874362795118482469989684865589317
1.86002 50792 21190 30718
Spiral of Theodorus [ 58]
∂
{\displaystyle \partial }
∑
n
=
1
∞
1
n
3
+
n
=
∑
n
=
1
∞
1
n
(
n
+
1
)
{\displaystyle \sum _{n=1}^{\infty }{\frac {1}((\sqrt {n^{3))}+{\sqrt {n))))=\sum _{n=1}^{\infty }{\frac {1}((\sqrt {n))(n+1)))}
Sum[n=1 to ∞] {1/(n^(3/2) +n^(1/2))}
OEIS : A226317
[1;1,6,6,1,15,11,5,1,1,1,1,5,3,3,3,2,1,1,2,19,...]
-460 to -399
1.86002507922119030718069591571714332
2.79128 78474 77920 00329
Nested radical S5
S
5
{\displaystyle S_{5))
21
+
1
2
=
5
+
5
+
5
+
5
+
5
+
⋯
{\displaystyle \displaystyle {\frac ((\sqrt {21))+1}{2))=\scriptstyle \,{\sqrt {5+{\sqrt {5+{\sqrt {5+{\sqrt {5+{\sqrt {5+\cdots ))))))))))\;}
=
1
+
5
−
5
−
5
−
5
−
5
−
⋯
{\displaystyle =1+\,\scriptstyle {\sqrt {5-{\sqrt {5-{\sqrt {5-{\sqrt {5-{\sqrt {5-\cdots ))))))))))\;}
(sqrt(21)+1)/2
A
A222134
[2;1,3,1,3,1,3,1,3,1,3,1,3,1,3,1,3,1,3,1,3,1,3,...] [2;1,3 ]
?
2.79128784747792000329402359686400424
0.70710 67811 86547 52440 +0.70710 67811 86547 524 i [ Mw 57]
Square root of i [ 59]
i
{\displaystyle {\sqrt {i))}
−
1
4
=
1
+
i
2
=
e
i
π
4
=
cos
(
π
4
)
+
i
sin
(
π
4
)
{\displaystyle {\sqrt[{4}]{-1))={\frac {1+i}{\sqrt {2))}=e^{\frac {i\pi }{4))=\cos \left({\frac {\pi }{4))\right)+i\sin \left({\frac {\pi }{4))\right)}
(1+i)/(sqrt 2)
C A
OEIS : A010503
[0;1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,..] = [0;1,2 ,...] [0;1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,..] i = [0;1,2 ,...] i
?
0.70710678118654752440084436210484903 + 0.70710678118654752440084436210484 i
0.80939 40205 40639 13071 [ Mw 58]
Alladi–Grinstead constant [ 60]
A
A
G
{\displaystyle ((\mathcal {A))_{AG))}
e
−
1
+
∑
k
=
2
∞
∑
n
=
1
∞
1
n
k
n
+
1
=
e
−
1
−
∑
k
=
2
∞
1
k
ln
(
1
−
1
k
)
{\displaystyle e^{-1+\sum \limits _{k=2}^{\infty }\sum \limits _{n=1}^{\infty }{\frac {1}{nk^{n+1))))=e^{-1-\sum \limits _{k=2}^{\infty }{\frac {1}{k))\ln \left(1-{\frac {1}{k))\right)))
e^{(sum[k=2 to ∞] |sum[n=1 to ∞] {1/(n k^(n+1))})-1}
OEIS : A085291
[0;1,4,4,17,4,3,2,5,3,1,1,1,1,6,1,1,2,1,22,...]
1977
0.80939402054063913071793188059409131
2.58498 17595 79253 21706 [ Mw 59]
Sierpiński's constant [ 61]
K
{\displaystyle {K))
π
(
2
γ
+
ln
4
π
3
Γ
(
1
4
)
4
)
=
π
(
2
γ
+
4
ln
Γ
(
3
4
)
−
ln
π
)
{\displaystyle \pi \left(2\gamma +\ln {\frac {4\pi ^{3)){\Gamma ({\tfrac {1}{4)))^{4))}\right)=\pi (2\gamma +4\ln \Gamma ({\tfrac {3}{4)))-\ln \pi )}
=
π
(
2
ln
2
+
3
ln
π
+
2
γ
−
4
ln
Γ
(
1
4
)
)
{\displaystyle =\pi \left(2\ln 2+3\ln \pi +2\gamma -4\ln \Gamma ({\tfrac {1}{4)))\right)}
-Pi Log[Pi]+2 Pi EulerGamma +4 Pi Log [Gamma[3/4]]
OEIS : A062089
[2;1,1,2,2,3,1,3,1,9,2,8,4,1,13,3,1,15,18,1,...]
1907
2.58498175957925321706589358738317116
1.73245 47146 00633 47358 [ Ow 1]
Reciprocal of the Euler–Mascheroni constant
1
γ
{\displaystyle {\frac {1}{\gamma ))}
(
∫
0
1
−
log
(
log
1
x
)
d
x
)
−
1
=
∑
n
=
1
∞
(
−
1
)
n
(
−
1
+
γ
)
n
{\displaystyle \left(\int _{0}^{1}-\log \left(\log {\frac {1}{x))\right)\,dx\right)^{-1}=\sum _{n=1}^{\infty }(-1)^{n}(-1+\gamma )^{n))
1/Integrate_ {x=0 to 1} -log(log(1/x))
OEIS : A098907
[1;1,2,1,2,1,4,3,13,5,1,1,8,1,2,4,1,1,40,1,11,...]
1.73245471460063347358302531586082968
1.43599 11241 76917 43235 [ Mw 60]
Lebesgue constant (interpolation) [ 62] [ 63]
L
1
{\displaystyle {L_{1))}
∏
i
=
0
j
≠
i
n
x
−
x
i
x
j
−
x
i
=
1
π
∫
0
π
⌊
sin
3
t
2
⌋
sin
t
2
d
t
=
1
3
+
2
3
π
{\displaystyle \prod _{\begin{smallmatrix}i=0\\j\neq i\end{smallmatrix))^{n}{\frac {x-x_{i)){x_{j}-x_{i))}={\frac {1}{\pi ))\int _{0}^{\pi }{\frac {\lfloor \sin {\frac {3t}{2))\rfloor }{\sin {\frac {t}{2))))\,dt={\frac {1}{3))+{\frac {2{\sqrt {3))}{\pi ))}
1/3 + 2*sqrt(3)/pi
T
OEIS : A226654
[1;2,3,2,2,6,1,1,1,1,4,1,7,1,1,1,2,1,3,1,2,1,1,...]
1902 ~
1.43599112417691743235598632995927221
3.24697 96037 17467 06105 [ Mw 61]
Silver root Tutte–Beraha constant [ 64]
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 {\frac {2\pi }{7))=\textstyle 2+{\frac {2+{\sqrt[{3}]{7+7{\sqrt[{3}]{7+7{\sqrt[{3}]{\,7+\cdots ))))))}{1+{\sqrt[{3}]{7+7{\sqrt[{3}]{7+7{\sqrt[{3}]{\,7+\cdots ))))))))}
2+2 cos(2Pi/7)
A
OEIS : A116425
[3;4,20,2,3,1,6,10,5,2,2,1,2,2,1,18,1,1,3,2,...]
3.24697960371746706105000976800847962
1.94359 64368 20759 20505 [ Mw 62]
Euler Totient constant [ 65] [ 66]
E
T
{\displaystyle ET}
∏
p
(
1
+
1
p
(
p
−
1
)
)
p
= primes
=
ζ
(
2
)
ζ
(
3
)
ζ
(
6
)
=
315
ζ
(
3
)
2
π
4
{\displaystyle {\underset {p{\text{= primes))}{\prod _{p}{\Big (}1+{\frac {1}{p(p-1))){\Big )))}={\frac {\zeta (2)\zeta (3)}{\zeta (6)))={\frac {315\zeta (3)}{2\pi ^{4))))
zeta(2)*zeta(3) /zeta(6)
OEIS : A082695
[1;1,16,1,2,1,2,3,1,1,3,2,1,8,1,1,1,1,1,1,1,32,...]
1750
1.94359643682075920505707036257476343
1.49534 87812 21220 54191
Fourth root of five [ 67]
5
4
{\displaystyle {\sqrt[{4}]{5))}
5
5
5
5
5
⋯
5
5
5
5
5
{\displaystyle {\sqrt[{5}]{5\,{\sqrt[{5}]{5\,{\sqrt[{5}]{5\,{\sqrt[{5}]{5\,{\sqrt[{5}]{5\,\cdots ))))))))))}
(5(5(5(5(5(5(5) ^1/5)^1/5)^1/5) ^1/5)^1/5)^1/5) ^1/5 ...
I
OEIS : A011003
[1;2,53,4,96,2,1,6,2,2,2,6,1,4,1,49,17,2,3,2,...]
1.49534878122122054191189899414091339
0.87228 40410 65627 97617 [ Mw 63]
Area of Ford circle [ 68]
A
C
F
{\displaystyle A_{CF))
∑
q
≥
1
∑
(
p
,
q
)
=
1
1
≤
p
<
q
π
(
1
2
q
2
)
2
=
π
4
ζ
(
3
)
ζ
(
4
)
=
45
2
ζ
(
3
)
π
3
ζ
(
)
= Riemann Zeta Function
{\displaystyle \sum _{q\geq 1}\sum _{(p,q)=1 \atop 1\leq p<q}\pi \left({\frac {1}{2q^{2))}\right)^{2}{\underset {\zeta (){\text{= Riemann Zeta Function))}{={\frac {\pi }{4)){\frac {\zeta (3)}{\zeta (4)))={\frac {45}{2)){\frac {\zeta (3)}{\pi ^{3))))))
pi Zeta(3) /(4 Zeta(4))
[0;1,6,1,4,1,7,5,36,3,29,1,1,10,3,2,8,1,1,1,3,...]
0.87228404106562797617519753217122587
1.08232 32337 11138 19151 [ Mw 64]
Zeta(4) [ 69]
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))}+...}
Sum[n=1 to ∞] {1/n^4}
T
OEIS : A013662
[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.[ 70]
R
2
{\displaystyle {R_{2))}
17
−
1
2
=
4
+
4
+
4
+
4
+
4
+
4
+
⋯
−
1
{\displaystyle {\frac ((\sqrt {17))-1}{2))=\,\scriptstyle {\sqrt {4+{\sqrt {4+{\sqrt {4+{\sqrt {4+{\sqrt {4+{\sqrt {4+\cdots ))))))))))))\,\,-1}
=
4
−
4
−
4
−
4
−
4
−
4
−
⋯
{\displaystyle =\,\scriptstyle {\sqrt {4-{\sqrt {4-{\sqrt {4-{\sqrt {4-{\sqrt {4-{\sqrt {4-\cdots ))))))))))))\textstyle }
(sqrt(17)-1)/2
A
OEIS : A222133
[1;1,1,3,1,1,3,1,1,3,1,1,3,1,1,3,1,1,3,1,1,3,1,...] [1;1,1,3 ]
1.56155281280883027491070492798703851
9.86960 44010 89358 61883
Pi Squared
π
2
{\displaystyle {\pi }^{2))
6
ζ
(
2
)
=
6
∑
n
=
1
∞
1
n
2
=
6
1
2
+
6
2
2
+
6
3
2
+
6
4
2
+
⋯
{\displaystyle 6\,\zeta (2)=6\sum _{n=1}^{\infty }{\frac {1}{n^{2))}={\frac {6}{1^{2))}+{\frac {6}{2^{2))}+{\frac {6}{3^{2))}+{\frac {6}{4^{2))}+\cdots }
6 Sum[n=1 to ∞] {1/n^2}
T
A002388
[9;1,6,1,2,47,1,8,1,1,2,2,1,1,8,3,1,10,5,1,3,...]
9.86960440108935861883449099987615114
1.32471 79572 44746 02596 [ Mw 65]
Plastic number [ 71]
ρ
{\displaystyle {\rho ))
1
+
1
+
1
+
⋯
3
3
3
=
1
2
+
23
108
3
+
1
2
−
23
108
3
{\displaystyle {\sqrt[{3}]{1+\!{\sqrt[{3}]{1+\!{\sqrt[{3}]{1+\cdots ))))))=\textstyle {\sqrt[{3}]((\frac {1}{2))+\!{\sqrt {\frac {23}{108))))}+\!{\sqrt[{3}]((\frac {1}{2))-\!{\sqrt {\frac {23}{108))))))
(1+(1+(1+(1+(1+(1 )^(1/3))^(1/3))^(1/3)) ^(1/3))^(1/3))^(1/3)
A
OEIS : A060006
[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 [ 72]
2
l
n
γ
{\displaystyle 2\,ln\,\gamma }
π
2
6
l
n
(
2
)
{\displaystyle {\frac {\pi ^{2)){6ln(2)))}
Pi^(2)/(6*ln(2))
T
OEIS : A174606
[2;2,1,2,8,57,9,32,1,1,2,1,2,1,2,1,2,1,3,2,...]
1936
2.37313822083125090564344595189447424
0.85073 61882 01867 26036 [ Mw 66]
Regular paperfolding sequence [ 73] [ 74]
P
f
{\displaystyle {P_{f))}
∑
n
=
0
∞
8
2
n
2
2
n
+
2
−
1
=
∑
n
=
0
∞
1
2
2
n
1
−
1
2
2
n
+
2
{\displaystyle \sum _{n=0}^{\infty }{\frac {8^{2^{n))}{2^{2^{n+2))-1))=\sum _{n=0}^{\infty }{\cfrac {\tfrac {1}{2^{2^{n)))){1-{\tfrac {1}{2^{2^{n+2))))))}
N[Sum[n=0 to ∞] {8^2^n/(2^2^ (n+2)-1)},37]
OEIS : A143347
[0;1,5,1,2,3,21,1,4,107,7,5,2,1,2,1,1,2,1,6,...]
0.85073618820186726036779776053206660
1.15636 26843 32269 71685 [ Mw 67]
Cubic recurrence constant [ 75] [ 76]
σ
3
{\displaystyle {\sigma _{3))}
∏
n
=
1
∞
n
3
−
n
=
1
2
3
⋯
3
3
3
=
1
1
/
3
2
1
/
9
3
1
/
27
⋯
{\displaystyle \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 }
prod[n=1 to ∞] {n ^(1/3)^n}
OEIS : A123852
[1;6,2,1,1,8,13,1,3,2,2,6,2,1,2,1,1,1,10,33,...]
1.15636268433226971685337032288736935
1.26185 95071 42914 87419 [ Mw 68]
Fractal dimension of the Koch snowflake [ 77]
C
k
{\displaystyle {C_{k))}
log
4
log
3
{\displaystyle {\frac {\log 4}{\log 3))}
log(4)/log(3)
I
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[ 78]
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,...]
6.58088599101792097085154240388648649
0.26149 72128 47642 78375 [ Mw 69]
Meissel-Mertens constant [ 79]
M
{\displaystyle {M))
lim
n
→
∞
(
∑
p
≤
n
1
p
−
ln
(
ln
(
n
)
)
)
=
γ
+
∑
p
(
ln
(
1
−
1
p
)
+
1
p
)
γ
:
Euler constant
,
p
:
prime
{\displaystyle \lim _{n\rightarrow \infty }\!\!\left(\sum _{p\leq n}{\frac {1}{p))\!-\ln(\ln(n))\!\right)\!\!={\underset {\!\!\!\!\gamma :\,{\text{Euler constant)),\,\,p:\,{\text{prime))}{\!\gamma \!+\!\!\sum _{p}\!\left(\!\ln \!\left(\!1\!-\!{\frac {1}{p))\!\right)\!\!+\!{\frac {1}{p))\!\right)))}
gamma+ Sum[n=1 to ∞] {ln(1-1/prime(n)) +1/prime(n)}
T ?
OEIS : A077761
[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 [ 80]
γ
{\displaystyle \gamma }
i
i
=
i
−
i
=
(
i
i
)
−
1
=
(
(
(
i
)
i
)
i
)
i
=
e
π
2
=
∑
n
=
0
∞
π
n
n
!
{\displaystyle {\sqrt[{i}]{i))=i^{-i}=(i^{i})^{-1}=(((i)^{i})^{i})^{i}=e^{\frac {\pi }{2))={\sqrt {\sum _{n=0}^{\infty }{\frac {\pi ^{n)){n!))))}
e^(π/2)
T
OEIS : A042972
[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 [ 81]
1
3
{\displaystyle {\sqrt[{3}]{1))}
{
1
−
1
2
+
3
2
i
−
1
2
−
3
2
i
.
{\displaystyle {\begin{cases}\ \ 1\\-{\frac {1}{2))+{\frac {\sqrt {3)){2))i\\-{\frac {1}{2))-{\frac {\sqrt {3)){2))i.\end{cases))}
1, E^(2i pi/3), E^(-2i pi/3)
C
OEIS : A010527
- [0,5] ± [0;1,6,2,6,2,6,2,6,2,6,2,6,2,6,2,6,2,6,2,...] i - [0,5] ± [0; 1, 6, 2 ] i
- 0.5 ± 0.8660254037844386467637231707529 i
0.11000 10000 00000 00000 0001 [ Mw 70]
Liouville number [ 82]
£
L
i
{\displaystyle {\text{£))_{Li))
∑
n
=
1
∞
1
10
n
!
=
1
10
1
!
+
1
10
2
!
+
1
10
3
!
+
1
10
4
!
+
⋯
{\displaystyle \sum _{n=1}^{\infty }{\frac {1}{10^{n!))}={\frac {1}{10^{1!))}+{\frac {1}{10^{2!))}+{\frac {1}{10^{3!))}+{\frac {1}{10^{4!))}+\cdots }
Sum[n=1 to ∞] {10^(-n!)}
T
OEIS : A012245
[1;9,1,999,10,9999999999999,1,9,999,1,9]
0.11000100000000000000000100...
0.06598 80358 45312 53707 [ Mw 71]
Lower limit of Tetration [ 83]
e
−
e
{\displaystyle {e}^{-e))
(
1
e
)
e
{\displaystyle \left({\frac {1}{e))\right)^{e))
1/(e^e)
OEIS : A073230
[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[ 84]
ϕ
3
{\displaystyle {\phi _{))_{3))
1
+
19
+
3
33
3
+
19
−
3
33
3
3
=
1
+
(
1
2
+
1
2
+
1
2
+
.
.
.
3
3
3
)
−
1
{\displaystyle \textstyle {\frac {1+{\sqrt[{3}]{19+3{\sqrt {33))))+{\sqrt[{3}]{19-3{\sqrt {33))))}{3))=\scriptstyle \,1+\left({\sqrt[{3}]((\tfrac {1}{2))+{\sqrt[{3}]((\tfrac {1}{2))+{\sqrt[{3}]((\tfrac {1}{2))+...))))))\right)^{-1))
(1/3)*(1+(19+3 *sqrt(33))^(1/3) +(19-3 *sqrt(33))^(1/3))
A
OEIS : A058265
[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 [ 85]
l
l
2
{\displaystyle {ll_{2))}
−
ln
(
ln
(
2
)
)
{\displaystyle -\ln(\ln(2))}
-ln(ln(2))
A074785
[0;2,1,2,1,2,6,1,6,6,2,2,2,1,12,1,8,1,1,3,1,...]
0.36651292058166432701243915823266947
36.46215 96072 07911 7709
Pi^pi [ 86]
π
π
{\displaystyle \pi ^{\pi ))
π
π
{\displaystyle \pi ^{\pi ))
pi^pi
OEIS : A073233
[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 [ 87]
2
+
ζ
(
1
2
)
{\displaystyle 2+\zeta ({\tfrac {1}{2)))}
2
−
(
1
+
2
)
∑
n
=
1
∞
(
−
1
)
n
+
1
n
=
γ
+
∑
n
=
1
∞
(
−
1
)
2
n
γ
n
2
n
n
!
{\displaystyle {2{-}(1{+}{\sqrt {2)))\sum _{n=1}^{\infty }{\frac {(-1)^{n+1)){\sqrt {n))))=\gamma +\sum _{n=1}^{\infty }{\frac {(-1)^{2n}\;\gamma _{n)){2^{n}n!))}
γ +N[ sum[n=1 to ∞] {((-1)^(2n) gamma_n) /(2^n n!)}]
2-OEIS : A059750
[0;1,1,5,1,4,6,1,1,2,6,1,1,2,1,1,1,37,3,2,1,...]
0.53964549119041318711050084748470198
15.15426 22414 79264 1897 [ Mw 72]
Exponential reiterated constant [ 88]
e
e
{\displaystyle e^{e))
∑
n
=
0
∞
e
n
n
!
=
lim
n
→
∞
(
1
+
n
n
)
n
−
n
(
1
+
n
)
1
+
n
{\displaystyle \sum _{n=0}^{\infty }{\frac {e^{n)){n!))=\lim _{n\to \infty }\left({\frac {1+n}{n))\right)^{n^{-n}(1+n)^{1+n))}
Sum[n=0 to ∞] {(e^n)/n!}
OEIS : A073226
[15;6,2,13,1,3,6,2,1,1,5,1,1,1,9,4,1,1,1,6,7,...]
15.1542622414792641897604302726299119
0.64624 54398 94813 30426 [ Mw 73]
Masser–Gramain constant [ 89]
C
{\displaystyle {C))
γ
β
(
1
)
+
β
′
(
1
)
=
π
(
−
ln
Γ
(
1
4
)
+
3
4
π
+
1
2
ln
2
+
1
2
γ
)
{\displaystyle \gamma {\beta }(1)\!+\!{\beta }'(1)\!=\pi \!\left(-\!\ln \Gamma ({\tfrac {1}{4)))+{\tfrac {3}{4))\pi +{\tfrac {1}{2))\ln 2+{\tfrac {1}{2))\gamma \right)}
=
π
(
−
ln
(
1
4
!
)
+
3
4
ln
π
−
3
2
ln
2
+
1
2
γ
)
{\displaystyle =\pi \!\left(-\!\ln({\tfrac {1}{4))!)+{\tfrac {3}{4))\ln \pi -{\tfrac {3}{2))\ln 2+{\tfrac {1}{2))\,\gamma \right)}
γ
=
Euler–Mascheroni constant
=
0.5772156649
…
{\displaystyle \scriptstyle \gamma ={\text{Euler–Mascheroni constant))=0.5772156649\ldots }
β
(
)
=
Beta function
,
Γ
(
)
=
Gamma function
{\displaystyle \scriptstyle \beta ()={\text{Beta function)),\quad \scriptstyle \Gamma ()={\text{Gamma function))}
Pi/4*(2*Gamma + 2*Log[2] + 3*Log[Pi]- 4 Log[Gamma[1/4]])
OEIS : A086057
[0;1,1,1,4,1,3,2,3,9,1,33,1,4,3,3,5,3,1,3,4,...]
0.64624543989481330426647339684579279
1.11072 07345 39591 56175 [ Mw 74]
The ratio of a square and circle circumscribed [ 90]
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))-{\cdots ))
sum[n=1 to ∞] {(-1)^(floor( (n-1)/2)) /(2n-1)}
T
OEIS : A093954
[1;9,31,1,1,17,2,3,3,2,3,1,1,2,2,1,4,9,1,3,...]
1.11072073453959156175397024751517342
1.45607 49485 82689 67139 [ Mw 75]
Backhouse's constant [ 91]
B
{\displaystyle {B))
lim
k
→
∞
|
q
k
+
1
q
k
|
where:
Q
(
x
)
=
1
P
(
x
)
=
∑
k
=
1
∞
q
k
x
k
{\displaystyle \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
)
=
∑
k
=
1
∞
p
k
x
k
p
k
prime
=
1
+
2
x
+
3
x
2
+
5
x
3
+
⋯
{\displaystyle P(x)=\sum _{k=1}^{\infty }{\underset {p_{k}{\text{ prime))}{p_{k}x^{k))}=1+2x+3x^{2}+5x^{3}+\cdots }
1/( FindRoot[0 == 1 + Sum[x^n Prime[n], {n, 10000}], {x, {1)))
OEIS : A072508
[1;2,5,5,4,1,1,18,1,1,1,1,1,2,13,3,1,2,4,16,...]
1995
1.45607494858268967139959535111654355
1.85193 70519 82466 17036 [ Mw 76]
Gibbs constant [ 92]
S
i
(
π
)
{\displaystyle {Si(\pi )))
Sin integral
∫
0
π
sin
t
t
d
t
=
∑
n
=
1
∞
(
−
1
)
n
−
1
π
2
n
−
1
(
2
n
−
1
)
(
2
n
−
1
)
!
{\displaystyle \int _{0}^{\pi }{\frac {\sin t}{t))\,dt=\sum \limits _{n=1}^{\infty }(-1)^{n-1}{\frac {\pi ^{2n-1)){(2n-1)(2n-1)!))}
=
π
−
π
3
3
⋅
3
!
+
π
5
5
⋅
5
!
−
π
7
7
⋅
7
!
+
⋯
{\displaystyle =\pi -{\frac {\pi ^{3)){3\cdot 3!))+{\frac {\pi ^{5)){5\cdot 5!))-{\frac {\pi ^{7)){7\cdot 7!))+\cdots }
SinIntegral[Pi]
OEIS : A036792
[1;1,5,1,3,15,1,5,3,2,7,2,1,62,1,3,110,1,39,...]
1.85193705198246617036105337015799136
0.23571 11317 19232 93137 [ Mw 77]
Copeland–Erdős constant [ 93]
C
C
E
{\displaystyle ((\mathcal {C))_{CE))}
∑
n
=
1
∞
p
n
10
n
+
∑
k
=
1
n
⌊
log
10
p
k
⌋
{\displaystyle \sum _{n=1}^{\infty }{\frac {p_{n)){10^{n+\sum \limits _{k=1}^{n}\lfloor \log _{10}{p_{k))\rfloor ))))
sum[n=1 to ∞] {prime(n) /(n+(10^ sum[k=1 to n]{floor (log_10 prime(k))}))}
A
OEIS : A033308
[0;4,4,8,16,18,5,1,1,1,1,7,1,1,6,2,9,58,1,3,...]
0.23571113171923293137414347535961677
1.52362 70862 02492 10627 [ Mw 78]
Fractal dimension of the boundary of the dragon curve [ 94]
C
d
{\displaystyle {C_{d))}
log
(
1
+
73
−
6
87
3
+
73
+
6
87
3
3
)
log
(
2
)
{\displaystyle {\frac {\log \left({\frac {1+{\sqrt[{3}]{73-6{\sqrt {87))))+{\sqrt[{3}]{73+6{\sqrt {87))))}{3))\right)}{\log(2)))}
(log((1+(73-6 sqrt(87))^1/3+ (73+6 sqrt(87))^1/3) /3))/ log(2)))
[1;1,1,10,12,2,1,149,1,1,1,3,11,1,3,17,4,1,...]
1.52362708620249210627768393595421662
1.78221 39781 91369 11177 [ Mw 79]
Grothendieck constant [ 95]
K
R
{\displaystyle {K_{R))}
π
2
log
(
1
+
2
)
{\displaystyle {\frac {\pi }{2\log(1+{\sqrt {2)))))}
pi/(2 log(1+sqrt(2)))
OEIS : A088367
[1;1,3,1,1,2,4,2,1,1,17,1,12,4,3,5,10,1,1,3,...]
1.78221397819136911177441345297254934
1.58496 25007 21156 18145 [ Mw 80]
Hausdorff dimension , Sierpinski triangle [ 96]
l
o
g
2
3
{\displaystyle {log_{2}3))
log
3
log
2
=
∑
n
=
0
∞
1
2
2
n
+
1
(
2
n
+
1
)
∑
n
=
0
∞
1
3
2
n
+
1
(
2
n
+
1
)
=
1
2
+
1
24
+
1
160
+
⋯
1
3
+
1
81
+
1
1215
+
⋯
{\displaystyle {\frac {\log 3}{\log 2))={\frac {\sum _{n=0}^{\infty }{\frac {1}{2^{2n+1}(2n+1)))}{\sum _{n=0}^{\infty }{\frac {1}{3^{2n+1}(2n+1)))))={\frac ((\frac {1}{2))+{\frac {1}{24))+{\frac {1}{160))+\cdots }((\frac {1}{3))+{\frac {1}{81))+{\frac {1}{1215))+\cdots ))}
( Sum[n=0 to ∞] {1/ (2^(2n+1) (2n+1))})/ (Sum[n=0 to ∞] {1/ (3^(2n+1) (2n+1))})
T
OEIS : A020857
[1;1,1,2,2,3,1,5,2,23,2,2,1,1,55,1,4,3,1,1,...]
1.58496250072115618145373894394781651
1.30637 78838 63080 69046 [ Mw 81]
Mills' constant [ 97]
θ
{\displaystyle {\theta ))
⌊
A
3
n
⌋
{\displaystyle \lfloor A^{3^{n))\rfloor }
Nest[ NextPrime[#^3] &, 2, 7]^(1/3^8)
OEIS : A051021
[1;3,3,1,3,1,2,1,2,1,4,2,35,21,1,4,4,1,1,3,2,...]
1947
1.30637788386308069046861449260260571
2.02988 32128 19307 25004 [ Mw 82]
Figure eight knot hyperbolic volume [ 98]
V
8
{\displaystyle {V_{8))}
2
3
∑
n
=
1
∞
1
n
(
2
n
n
)
∑
k
=
n
2
n
−
1
1
k
=
6
∫
0
π
/
3
log
(
1
2
sin
t
)
d
t
=
{\displaystyle 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=}
3
9
∑
n
=
0
∞
(
−
1
)
n
27
n
{
18
(
6
n
+
1
)
2
−
18
(
6
n
+
2
)
2
−
24
(
6
n
+
3
)
2
−
6
(
6
n
+
4
)
2
+
2
(
6
n
+
5
)
2
}
{\displaystyle \scriptstyle {\frac {\sqrt {3)){9))\,\sum \limits _{n=0}^{\infty }{\frac {(-1)^{n)){27^{n))}\,\left\{\!{\frac {18}{(6n+1)^{2))}-{\frac {18}{(6n+2)^{2))}-{\frac {24}{(6n+3)^{2))}-{\frac {6}{(6n+4)^{2))}+{\frac {2}{(6n+5)^{2))}\!\right\))
6 integral[0 to pi/3] {log(1/(2 sin (n)))}
OEIS : A091518
[2;33,2,6,2,1,2,2,5,1,1,7,1,1,1,113,1,4,5,1,...]
2.02988321281930725004240510854904057
262 53741 26407 68743 .99999 99999 99250 073 [ Mw 83]
Hermite–Ramanujan constant [ 99]
R
{\displaystyle {R))
e
π
163
{\displaystyle e^{\pi {\sqrt {163))))
e^(π sqrt(163))
T
OEIS : A060295
[262537412640768743;1,1333462407511,1,8,1,1,5,...]
1859
262537412640768743.999999999999250073
1.74540 56624 07346 86349 [ Mw 84]
Khinchin harmonic mean [ 100]
K
−
1
{\displaystyle {K_{-1))}
log
2
∑
n
=
1
∞
1
n
log
(
1
+
1
n
(
n
+
2
)
)
=
lim
n
→
∞
n
1
a
1
+
1
a
2
+
⋯
+
1
a
n
{\displaystyle {\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 ]
(log 2)/ (sum[n=1 to ∞] {1/n log(1+ 1/(n(n+2))}
OEIS : A087491
[1;1,2,1,12,1,5,1,5,13,2,13,2,1,9,1,6,1,3,1,...]
1.74540566240734686349459630968366106
1.64872 12707 00128 14684 [ Ow 2]
Square root of the number e [ 101]
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
OEIS : 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
1.64872127070012814684865078781416357
1.01734 30619 84449 13971 [ Mw 85]
Zeta(6) [ 102]
ζ
(
6
)
{\displaystyle \zeta (6)}
π
6
945
=
∏
n
=
1
∞
1
1
−
p
n
−
6
p
n
:
prime
=
1
1
−
2
−
6
⋅
1
1
−
3
−
6
⋅
1
1
−
5
−
6
⋯
{\displaystyle {\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 }
Prod[n=1 to ∞] {1/(1-ithprime (n)^-6)}
T
OEIS : A013664
[1;57,1,1,1,15,1,6,3,61,1,5,3,1,6,1,3,3,6,1,...]
1.01734306198444913971451792979092052
0.10841 01512 23111 36151 [ Mw 86]
Trott constant [ 103]
T
1
{\displaystyle \mathrm {T} _{1))
[
1
,
0
,
8
,
4
,
1
,
0
,
1
,
5
,
1
,
2
,
2
,
3
,
1
,
1
,
1
,
3
,
6
,
.
.
.
]
{\displaystyle \textstyle [1,0,8,4,1,0,1,5,1,2,2,3,1,1,1,3,6,...]}
1
1
+
1
0
+
1
8
+
1
4
+
1
1
+
1
0
+
1
/
⋯
{\displaystyle {\tfrac {1}{1+{\tfrac {1}{0+{\tfrac {1}{8+{\tfrac {1}{4+{\tfrac {1}{1+{\tfrac {1}{0+1{/\cdots ))))))))))))))
Trott Constant
OEIS : A039662
[0;9,4,2,5,1,2,2,3,1,1,1,3,6,1,5,1,1,2,...]
0.10841015122311136151129081140641509
0.00787 49969 97812 3844 [ Mw 87]
Chaitin constant [ 104]
Ω
{\displaystyle \Omega }
∑
p
∈
P
2
−
|
p
|
|
p
|
:
Size in bits of program
p
P
:
Domain of all programs that stop.
p
:
Halted program
{\displaystyle \sum _{p\in P}2^{-|p|}{\overset {p:{\text{ Halted program))}{\underset {P:{\text{ Domain of all programs that stop.))}{\scriptstyle {|p|}:{\text{Size in bits of program ))p))))
See also: Halting problem
T
OEIS : A100264
[0; 126, 1, 62, 5, 5, 3, 3, 21, 1, 4, 1]
1975
0.0078749969978123844
0.83462 68416 74073 18628 [ Mw 88]
Gauss constant [ 105]
G
{\displaystyle {G))
1
a
g
m
(
1
,
2
)
=
4
2
(
1
4
!
)
2
π
3
/
2
=
2
π
∫
0
1
d
x
1
−
x
4
{\displaystyle {\frac {1}{\mathrm {agm} (1,{\sqrt {2)))))={\frac {4{\sqrt {2))\,({\tfrac {1}{4))!)^{2)){\pi ^{3/2))}={\frac {2}{\pi ))\int _{0}^{1}{\frac {dx}{\sqrt {1-x^{4))))}
AGM = Arithmetic–geometric mean
(4 sqrt(2)((1/4)!)^2) /pi^(3/2)
T
OEIS : A014549
[0;1,5,21,3,4,14,1,1,1,1,1,3,1,15,1,3,7,1,...]
0.83462684167407318628142973279904680
1.45136 92348 83381 05028 [ Mw 89]
Ramanujan–Soldner constant [ 106] [ 107]
μ
{\displaystyle {\mu ))
l
i
(
x
)
=
∫
0
x
d
t
ln
t
=
0
.
.
.
.
.
.
{\displaystyle \mathrm {li} (x)=\int \limits _{0}^{x}{\frac {dt}{\ln t))=0{\color {White}{......))}
li = Logarithmic integral
l
i
(
x
)
=
E
i
(
ln
x
)
.
.
.
.
.
.
.
.
{\displaystyle \mathrm {li} (x)\;=\;\mathrm {Ei} (\ln {x}){\color {White}{........))}
Ei = Exponential integral
FindRoot[li(x) = 0]
I
OEIS : A070769
[1;2,4,1,1,1,3,1,1,1,2,47,2,4,1,12,1,1,2,2,1,...]
1792 to 1809
1.45136923488338105028396848589202744
0.64341 05462 88338 02618 [ Mw 90]
Cahen's constant [ 108]
ξ
2
{\displaystyle \xi _{2))
∑
k
=
1
∞
(
−
1
)
k
s
k
−
1
=
1
1
−
1
2
+
1
6
−
1
42
+
1
1806
±
⋯
{\displaystyle \sum _{k=1}^{\infty }{\frac {(-1)^{k)){s_{k}-1))={\frac {1}{1))-{\frac {1}{2))+{\frac {1}{6))-{\frac {1}{42))+{\frac {1}{1806)){\,\pm \cdots ))
Where sk is the kth term of Sylvester's sequence 2, 3, 7, 43, 1807, ...
Defined as:
S
0
=
2
,
S
k
=
1
+
∏
n
=
0
k
−
1
S
n
for
k
>
0
{\displaystyle \,\,S_{0}=\,2,\,\,S_{k}=\,1+\prod \limits _{n=0}^{k-1}S_{n}{\text{ for))\;k>0}
T
OEIS : A080130
[0; 1, 1, 1, 4, 9, 196, 16641, 639988804, ...]
1891
0.64341054628833802618225430775756476
1.41421 35623 73095 04880 [ Mw 91]
Square root of 2 , Pythagoras constant.[ 109]
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 }\!\left(1\!+\!{\frac {(-1)^{n+1)){2n-1))\right)\!=\!\left(1\!+\!{\frac {1}{1))\right)\!\left(1\!-\!{\frac {1}{3))\right)\!\left(1\!+\!{\frac {1}{5))\right)\cdots }
prod[n=1 to ∞] {1+(-1)^(n+1) /(2n-1)}
A
OEIS : A002193
[1;2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,...] = [1;2 ...]
1.41421356237309504880168872420969808
1.77245 38509 05516 02729 [ Mw 92]
Carlson–Levin constant [ 110]
Γ
(
1
2
)
{\displaystyle {\Gamma }({\tfrac {1}{2)))}
π
=
(
−
1
2
)
!
=
∫
−
∞
∞
1
e
x
2
d
x
=
∫
0
1
1
−
ln
x
d
x
{\displaystyle {\sqrt {\pi ))=\left(-{\frac {1}{2))\right)!=\int _{-\infty }^{\infty }{\frac {1}{e^{x^{2))))\,dx=\int _{0}^{1}{\frac {1}{\sqrt {-\ln x))}\,dx}
sqrt (pi)
T
OEIS : A002161
[1;1,3,2,1,1,6,1,28,13,1,1,2,18,1,1,1,83,1,...]
1.77245385090551602729816748334114518
1.05946 30943 59295 26456 [ Ow 3]
Musical interval between each half tone [ 111] [ 112]
2
12
{\displaystyle {\sqrt[{12}]{2))}
440
H
z
.
2
1
12
2
2
12
2
3
12
2
4
12
2
5
12
2
6
12
2
7
12
2
8
12
2
9
12
2
10
12
2
11
12
2
{\displaystyle \scriptstyle 440\,Hz.\textstyle 2^{\frac {1}{12))\,2^{\frac {2}{12))\,2^{\frac {3}{12))\,2^{\frac {4}{12))\,2^{\frac {5}{12))\,2^{\frac {6}{12))\,2^{\frac {7}{12))\,2^{\frac {8}{12))\,2^{\frac {9}{12))\,2^{\frac {10}{12))\,2^{\frac {11}{12))\,2}
.
.
.
D
o
1
D
o
#
R
e
R
e
#
M
i
F
a
F
a
#
S
o
l
S
o
l
#
L
a
L
a
#
S
i
D
o
2
{\displaystyle \scriptstyle {\color {white}...\color {black}Do_{1}\;\;Do\#\;\,Re\;\,Re\#\;\,Mi\;\;Fa\;\;Fa\#\;Sol\;\,Sol\#\,La\;\;La\#\;\;Si\;\,Do_{2))}
.
.
.
.
C
1
C
#
D
D
#
E
F
F
#
G
G
#
A
A
#
B
C
2
{\displaystyle \scriptstyle {\color {white}....\color {black}C_{1}\;\;\;\;C\#\;\;\;\,D\;\;\;D\#\;\;\,E\;\;\;\;\,F\;\;\;\,F\#\;\;\;G\;\;\;\;G\#\;\;\;A\;\;\;\,A\#\;\;\;\,B\;\;\;C_{2))}
2^(1/12)
A
OEIS : A010774
[1;16,1,4,2,7,1,1,2,2,7,4,1,2,1,60,1,3,1,2,...]
1.05946309435929526456182529494634170
1.01494 16064 09653 62502 [ Mw 93]
Gieseking constant [ 113]
π
ln
β
{\displaystyle {\pi \ln \beta ))
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 \cdots \right)}
.
sqrt(3)*3/4 *(1 -Sum[n=0 to ∞] {1/((3n+2)^2)} +Sum[n=1 to ∞] {1/((3n+1)^2)})
OEIS : A143298
[1;66,1,12,1,2,1,4,2,1,3,3,1,4,1,56,2,2,11,...]
1912
1.01494160640965362502120255427452028
2.62205 75542 92119 81046 [ Mw 94]
Lemniscate constant [ 114]
ϖ
{\displaystyle {\varpi ))
π
G
=
4
2
π
Γ
(
5
4
)
2
=
1
4
2
π
Γ
(
1
4
)
2
=
4
2
π
(
1
4
!
)
2
{\displaystyle \pi \,{G}=4{\sqrt {\tfrac {2}{\pi ))}\,\Gamma {\left({\tfrac {5}{4))\right)^{2))={\tfrac {1}{4)){\sqrt {\tfrac {2}{\pi ))}\,\Gamma {\left({\tfrac {1}{4))\right)^{2))=4{\sqrt {\tfrac {2}{\pi ))}\left({\tfrac {1}{4))!\right)^{2))
4 sqrt(2/pi) ((1/4)!)^2
T
OEIS : A062539
[2;1,1,1,1,1,4,1,2,5,1,1,1,14,9,2,6,2,9,4,1,...]
1798
2.62205755429211981046483958989111941
1.28242 71291 00622 63687 [ Mw 95]
Glaisher–Kinkelin constant [ 115]
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^(1/12-zeta´{-1})
T ?
OEIS : A074962
[1;3,1,1,5,1,1,1,3,12,4,1,271,1,1,2,7,1,35,...]
1.28242712910062263687534256886979172
-4.22745 35333 76265 408 [ Mw 96]
Digamma (1/4) [ 116]
ψ
(
1
4
)
{\displaystyle {\psi }({\tfrac {1}{4)))}
−
γ
−
π
2
−
3
ln
2
=
−
γ
+
∑
n
=
0
∞
(
1
n
+
1
−
1
n
+
1
4
)
{\displaystyle -\gamma -{\frac {\pi }{2))-3\ln {2}=-\gamma +\sum _{n=0}^{\infty }\left({\frac {1}{n+1))-{\frac {1}{n+{\tfrac {1}{4))))\right)}
-EulerGamma -\pi/2 -3 log 2
OEIS : A020777
-[4;4,2,1,1,10,1,5,9,11,1,22,1,1,14,1,2,1,4,...]
-4.2274535333762654080895301460966835
0.28674 74284 34478 73410 [ Mw 97]
Strongly Carefree constant [ 117]
K
2
{\displaystyle K_{2))
∏
n
=
1
∞
(
1
−
3
p
n
−
2
p
n
3
)
p
n
:
prime
=
6
π
2
∏
n
=
1
∞
(
1
−
1
p
n
(
p
n
+
1
)
)
p
n
:
prime
{\displaystyle \prod _{n=1}^{\infty }{\underset {p_{n}:{\text{ prime))}{\left(1-{\frac {3p_{n}-2}((p_{n))^{3))}\right)))={\frac {6}{\pi ^{2))}\prod _{n=1}^{\infty }{\underset {p_{n}:{\text{ prime))}{\left(1-{\frac {1}{p_{n}(p_{n}+1)))\right)))}
N[ prod[k=1 to ∞] {1-(3*prime(k)-2) /(prime(k)^3)}]
OEIS : A065473
[0;3,2,19,3,12,1,5,1,5,1,5,2,1,1,1,1,1,3,7,...]
0.28674742843447873410789271278983845
1.78107 24179 90197 98523 [ Mw 98]
Exp.gamma, Barnes G-function [ 118]
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}\cdots }
Prod[n=1 to ∞] {e^(1/n)} /{1 + 1/n}
OEIS : A073004
[1;1,3,1,1,3,5,4,1,1,2,2,1,7,9,1,16,1,1,1,2,...]
1.78107241799019798523650410310717954
3.62560 99082 21908 31193 [ Mw 99]
Gamma(1/4)[ 119]
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
OEIS : A068466
[3;1,1,1,2,25,4,9,1,1,8,4,1,6,1,1,19,1,1,4,1,...]
1729
3.62560990822190831193068515586767200
1.66168 79496 33594 12129 [ Mw 100]
Somos' quadratic recurrence constant [ 120]
σ
{\displaystyle {\sigma ))
∏
n
=
1
∞
n
1
/
2
n
=
1
2
3
⋯
=
1
1
/
2
2
1
/
4
3
1
/
8
⋯
{\displaystyle \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 }
prod[n=1 to ∞] {n ^(1/2)^n}
T ?
OEIS : A065481
[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 [ 121]
θ
m
{\displaystyle {\theta _{m))}
arctan
(
2
)
=
arccos
(
1
3
)
≈
54.7356
∘
{\displaystyle \arctan \left({\sqrt {2))\right)=\arccos \left({\sqrt {\tfrac {1}{3))}\right)\approx \textstyle {54.7356}^{\circ ))
arctan(sqrt(2))
I
OEIS : A195696
[0;1,21,2,1,1,1,2,1,2,2,4,1,2,9,1,2,1,1,1,3,...]
0.95531661812450927816385710251575775
0.74759 79202 53411 43517 [ Mw 101]
Rényi's Parking Constant [ 122]
m
{\displaystyle {m))
∫
0
∞
e
x
p
(
−
2
∫
0
x
1
−
e
−
y
y
d
y
)
d
x
=
e
−
2
γ
∫
0
∞
e
−
2
Γ
(
0
,
n
)
n
2
{\displaystyle \int \limits _{0}^{\infty }exp\left(\!-2\int \limits _{0}^{x}{\frac {1-e^{-y)){y))dy\right)\!dx={e^{-2\gamma ))\int \limits _{0}^{\infty }{\frac {e^{-2\Gamma (0,n))){n^{2))))
[e^(-2*Gamma)] * Int{n,0,∞}[ e^(- 2 *Gamma(0,n)) /n^2]
OEIS : A050996
[0;1,2,1,25,3,1,2,1,1,12,1,2,1,1,3,1,2,1,43,...]
0.74759792025341143517873094383017817
1.44466 78610 09766 13365 [ Mw 102]
Steiner number, Iterated exponential Constant [ 123]
e
e
{\displaystyle {\sqrt[{e}]{e))}
e
1
e
.
.
.
.
.
.
.
.
.
.
.
{\displaystyle e^{\frac {1}{e)){\color {White}{...........))}
= Upper Limit of Tetration
e^(1/e)
T
OEIS : A073229
[1;2,4,55,27,1,1,16,9,3,2,8,3,2,1,1,4,1,9,...]
1.44466786100976613365833910859643022
0.69220 06275 55346 35386 [ Mw 103]
Minimum value of función ƒ (x) = xx [ 124]
(
1
e
)
1
e
{\displaystyle {\left({\frac {1}{e))\right)}^{\frac {1}{e))}
e
−
1
e
.
.
.
.
.
.
.
.
.
.
{\displaystyle {e}^{-{\frac {1}{e))}{\color {White}{..........))}
= Inverse Steiner Number
e^(-1/e)
OEIS : A072364
[0;1,2,4,55,27,1,1,16,9,3,2,8,3,2,1,1,4,1,9,...]
0.69220062755534635386542199718278976
0.34053 73295 50999 14282 [ Mw 104]
Pólya Random walk constant [ 125]
p
(
3
)
{\displaystyle {p(3)))
1
−
(
3
(
2
π
)
3
∫
−
π
π
∫
−
π
π
∫
−
π
π
d
x
d
y
d
z
3
−
cos
x
−
cos
y
−
cos
z
)
−
1
{\displaystyle 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
2
3
π
3
(
Γ
(
1
24
)
Γ
(
5
24
)
Γ
(
7
24
)
Γ
(
11
24
)
)
−
1
{\displaystyle =1-16{\sqrt {\tfrac {2}{3))}\;\pi ^{3}\left(\Gamma ({\tfrac {1}{24)))\Gamma ({\tfrac {5}{24)))\Gamma ({\tfrac {7}{24)))\Gamma ({\tfrac {11}{24)))\right)^{-1))
1-16*Sqrt[2/3]*Pi^3 /(Gamma[1/24] *Gamma[5/24] *Gamma[7/24] *Gamma[11/24])
OEIS : A086230
[0;2,1,14,1,3,8,1,5,2,7,1,12,1,5,59,1,1,1,3,...]
0.34053732955099914282627318443290289
0.54325 89653 42976 70695 [ Mw 105]
Bloch–Landau constant [ 126]
L
{\displaystyle {L))
=
Γ
(
1
3
)
Γ
(
5
6
)
Γ
(
1
6
)
=
(
−
2
3
)
!
(
−
1
+
5
6
)
!
(
−
1
+
1
6
)
!
{\displaystyle ={\frac {\Gamma ({\tfrac {1}{3)))\;\Gamma ({\tfrac {5}{6)))}{\Gamma ({\tfrac {1}{6)))))={\frac {(-{\tfrac {2}{3)))!\;(-1+{\tfrac {5}{6)))!}{(-1+{\tfrac {1}{6)))!))}
gamma(1/3) *gamma(5/6) /gamma(1/6)
OEIS : A081760
[0;1,1,5,3,1,1,2,1,1,6,3,1,8,11,2,1,1,27,4,...]
1929
0.54325896534297670695272829530061323
0.18785 96424 62067 12024 [ Mw 106] [ Ow 4]
MRB Constant , Marvin Ray Burns [ 127] [ 128] [ 129]
C
M
R
B
{\displaystyle C_((}_{MRB))}
∑
n
=
1
∞
(
−
1
)
n
(
n
1
/
n
−
1
)
=
−
1
1
+
2
2
−
3
3
+
⋯
{\displaystyle \sum _{n=1}^{\infty }(-1)^{n}(n^{1/n}-1)=-{\sqrt[{1}]{1))+{\sqrt[{2}]{2))-{\sqrt[{3}]{3))+\cdots }
Sum[n=1 to ∞] {(-1)^n (n^(1/n)-1)}
OEIS : A037077
[0;5,3,10,1,1,4,1,1,1,1,9,1,1,12,2,17,2,2,1,...]
1999
0.18785964246206712024851793405427323
1.27323 95447 35162 68615
Ramanujan–Forsyth series[ 130]
4
π
{\displaystyle {\frac {4}{\pi ))}
∑
n
=
0
∞
(
(
2
n
−
3
)
!
!
(
2
n
)
!
!
)
2
=
1
+
(
1
2
)
2
+
(
1
2
⋅
4
)
2
+
(
1
⋅
3
2
⋅
4
⋅
6
)
2
+
⋯
{\displaystyle \displaystyle \sum \limits _{n=0}^{\infty }\textstyle \left({\frac {(2n-3)!!}{(2n)!!))\right)^{2}={1\!+\!\left({\frac {1}{2))\right)^{2}\!+\!\left({\frac {1}{2\cdot 4))\right)^{2}\!+\!\left({\frac {1\cdot 3}{2\cdot 4\cdot 6))\right)^{2}+\cdots ))
Sum[n=0 to ∞] {[(2n-3)!! /(2n)!!]^2}
I
OEIS : A088538
[1;3,1,1,1,15,2,72,1,9,1,17,1,2,1,5,1,1,10,...]
1.27323954473516268615107010698011489
1.46707 80794 33975 47289 [ Mw 107]
Porter Constant[ 131]
C
{\displaystyle {C))
6
ln
2
π
2
(
3
ln
2
+
4
γ
−
24
π
2
ζ
′
(
2
)
−
2
)
−
1
2
{\displaystyle {\frac {6\ln 2}{\pi ^{2))}\left(3\ln 2+4\,\gamma -{\frac {24}{\pi ^{2))}\,\zeta '(2)-2\right)-{\frac {1}{2))}
γ
= Euler–Mascheroni Constant
=
0.5772156649
…
{\displaystyle \scriptstyle \gamma \,{\text{= Euler–Mascheroni Constant))=0.5772156649\ldots }
ζ
′
(
2
)
= Derivative of
ζ
(
2
)
=
−
∑
n
=
2
∞
ln
n
n
2
=
−
0.9375482543
…
{\displaystyle \scriptstyle \zeta '(2)\,{\text{= Derivative of ))\zeta (2)=-\sum \limits _{n=2}^{\infty }{\frac {\ln n}{n^{2))}=-0.9375482543\ldots }
6*ln2/pi^2(3*ln2+ 4 EulerGamma- WeierstrassZeta'(2) *24/pi^2-2)-1/2
OEIS : A086237
[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 [ Mw 108]
Feigenbaum constant δ [ 132]
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
)
or
x
n
+
1
=
a
sin
(
x
n
)
{\displaystyle \scriptstyle x_{n+1}=\,ax_{n}(1-x_{n})\quad {\text{or))\quad x_{n+1}=\,a\sin(x_{n})}
T
OEIS : A006890
[4;1,2,43,2,163,2,3,1,1,2,5,1,2,3,80,2,5,...]
1975
4.66920160910299067185320382046620161
2.50290 78750 95892 82228 [ Mw 109]
Feigenbaum constant α[ 133]
α
{\displaystyle \alpha }
lim
n
→
∞
d
n
d
n
+
1
{\displaystyle \lim _{n\to \infty }{\frac {d_{n)){d_{n+1))))
T ?
OEIS : A006891
[2;1,1,85,2,8,1,10,16,3,8,9,2,1,40,1,2,3,...]
1979
2.50290787509589282228390287321821578
0.62432 99885 43550 87099 [ Mw 110]
Golomb–Dickman constant [ 134]
λ
{\displaystyle {\lambda ))
∫
0
∞
f
(
x
)
x
2
d
x
Para
x
>
2
=
∫
0
1
e
Li
(
n
)
d
n
Li: Logarithmic integral
{\displaystyle \int \limits _{0}^{\infty }{\underset ((\text{Para ))x>2}((\frac {f(x)}{x^{2))}\,dx))=\int \limits _{0}^{1}e^{\operatorname {Li} (n)}dn\quad \scriptstyle {\text{Li: Logarithmic integral))}
N[Int{n,0,1}[e^Li(n)],34]
OEIS : A084945
[0;1,1,1,1,1,22,1,2,3,1,1,11,1,1,2,22,2,6,1,...]
1930 & 1964
0.62432998854355087099293638310083724
23.14069 26327 79269 0057 [ Mw 111]
Gelfond constant [ 135]
e
π
{\displaystyle {e}^{\pi ))
(
−
1
)
−
i
=
i
−
2
i
=
∑
n
=
0
∞
π
n
n
!
=
π
1
1
+
π
2
2
!
+
π
3
3
!
+
⋯
{\displaystyle (-1)^{-i}=i^{-2i}=\sum _{n=0}^{\infty }{\frac {\pi ^{n)){n!))={\frac {\pi ^{1)){1))+{\frac {\pi ^{2)){2!))+{\frac {\pi ^{3)){3!))+\cdots }
Sum[n=0 to ∞] {(pi^n)/n!}
T
OEIS : A039661
[23;7,9,3,1,1,591,2,9,1,2,34,1,16,1,30,1,...]
23.1406926327792690057290863679485474
7.38905 60989 30650 22723
Conic constant , Schwarzschild constant [ 136]
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!))+\cdots }
Sum[n=0 to ∞] {2^n/n!}
OEIS : A072334
[7;2,1,1,3,18,5,1,1,6,30,8,1,1,9,42,11,1,...] = [7,2,1,1,n,4*n+6,n+2 ], n = 3, 6, 9, etc.
7.38905609893065022723042746057500781
0.35323 63718 54995 98454 [ Mw 112]
Hafner–Sarnak–McCurley constant (1) [ 137]
σ
{\displaystyle {\sigma ))
∏
k
=
1
∞
{
1
−
[
1
−
∏
j
=
1
n
(
1
−
p
k
−
j
)
]
2
p
k
:
prime
}
{\displaystyle \prod _{k=1}^{\infty }\left\{1-[1-\prod _{j=1}^{n}{\underset {p_{k}:{\text{ prime))}{(1-p_{k}^{-j})]^{2))}\right\))
prod[k=1 to ∞] {1-(1-prod[j=1 to n] {1-ithprime(k)^-j})^2}
OEIS : A085849
[0;2,1,4,1,10,1,8,1,4,1,2,1,2,1,2,6,1,1,1,3,...]
1993
0.35323637185499598454351655043268201
0.60792 71018 54026 62866 [ Mw 113]
Hafner–Sarnak–McCurley constant (2) [ 138]
1
ζ
(
2
)
{\displaystyle {\frac {1}{\zeta (2)))}
6
π
2
=
∏
n
=
0
∞
(
1
−
1
p
n
2
)
p
n
:
prime
=
(
1
−
1
2
2
)
(
1
−
1
3
2
)
(
1
−
1
5
2
)
⋯
{\displaystyle {\frac {6}{\pi ^{2))}=\prod _{n=0}^{\infty }{\underset {p_{n}:{\text{ prime))}{\!\left(\!1-{\frac {1}((p_{n))^{2))}\!\right)))\!=\!\textstyle \left(1\!-\!{\frac {1}{2^{2))}\right)\!\left(1\!-\!{\frac {1}{3^{2))}\right)\!\left(1\!-\!{\frac {1}{5^{2))}\right)\cdots }
Prod{n=1 to ∞} (1-1/ithprime(n)^2)
T
OEIS : A059956
[0;1,1,1,1,4,2,4,7,1,4,2,3,4,10,1,2,1,1,1,...]
0.60792710185402662866327677925836583
0.12345 67891 01112 13141 [ Mw 114]
Champernowne constant [ 139]
C
10
{\displaystyle C_{10))
∑
n
=
1
∞
∑
k
=
10
n
−
1
10
n
−
1
k
10
k
n
−
9
∑
j
=
0
n
−
1
10
j
(
n
−
j
−
1
)
{\displaystyle \sum _{n=1}^{\infty }\;\sum _{k=10^{n-1))^{10^{n}-1}{\frac {k}{10^{kn-9\sum _{j=0}^{n-1}10^{j}(n-j-1)))))
T
OEIS : A033307
[0;8,9,1,149083,1,1,1,4,1,1,1,3,4,1,1,1,15,...]
1933
0.12345678910111213141516171819202123
0.76422 36535 89220 66299 [ Mw 115]
Landau-Ramanujan constant [ 140]
K
{\displaystyle K}
1
2
∏
p
≡
3
mod
4
(
1
−
1
p
2
)
−
1
2
p
:
prime
=
π
4
∏
p
≡
1
mod
4
(
1
−
1
p
2
)
1
2
p
:
prime
{\displaystyle {\frac {1}{\sqrt {2))}\prod _{p\equiv 3\!\!\!\!\!\mod \!4}\!\!{\underset {\!\!\!\!\!\!\!\!p:{\text{ prime))}{\left(1-{\frac {1}{p^{2))}\right)^{-{\frac {1}{2))))}\!\!={\frac {\pi }{4))\prod _{p\equiv 1\!\!\!\!\!\mod \!4}\!\!{\underset {\!\!\!\!p:{\text{ prime))}{\left(1-{\frac {1}{p^{2))}\right)^{\frac {1}{2))))}
T ?
OEIS : A064533
[0;1,3,4,6,1,15,1,2,2,3,1,23,3,1,1,3,1,1,6,4,...]
0.76422365358922066299069873125009232
1.92878 00... [ Mw 116]
Wright constant [ 141]
ω
{\displaystyle {\omega ))
⌊
2
2
2
⋅
⋅
2
ω
⌋
= primes:
⌊
2
ω
⌋
=3,
⌊
2
2
ω
⌋
=13,
⌊
2
2
2
ω
⌋
=
16381
,
…
{\displaystyle \left\lfloor 2^{2^{2^{\cdot ^{\cdot ^{2^{\omega ))))))\!\right\rfloor \scriptstyle {\text{= primes:))\displaystyle \left\lfloor 2^{\omega }\right\rfloor \scriptstyle {\text{=3,))\displaystyle \left\lfloor 2^{2^{\omega ))\right\rfloor \scriptstyle {\text{=13,))\displaystyle \left\lfloor 2^{2^{2^{\omega ))}\right\rfloor \scriptstyle =16381,\ldots }
OEIS : A086238
[1; 1, 13, 24, 2, 1, 1, 3, 1, 1, 3]
1.9287800...
2.71828 18284 59045 23536 [ Mw 117]
Number e , Euler's number [ 142]
e
{\displaystyle {e))
lim
n
→
∞
(
1
+
1
n
)
n
=
∑
n
=
0
∞
1
n
!
=
1
0
!
+
1
1
+
1
2
!
+
1
3
!
+
⋯
{\displaystyle \!\lim _{n\to \infty }\!\left(\!1\!+\!{\frac {1}{n))\right)^{n}\!=\!\sum _{n=0}^{\infty }{\frac {1}{n!))={\frac {1}{0!))+{\frac {1}{1))+{\frac {1}{2!))+{\frac {1}{3!))+\textstyle \cdots }
Sum[n=0 to ∞] {1/n!}
T
OEIS : A001113
[2;1,2,1,1,4,1,1,6,1,1,8,1,1,10,1,1,12,1,...] = [2;1,2p,1 ], p∈ℕ
2.71828182845904523536028747135266250
0.36787 94411 71442 32159 [ Mw 118]
Inverse of Number e [ 143]
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!))+\cdots }
Sum[n=2 to ∞] {(-1)^n/n!}
T
OEIS : A068985
[0;2,1,1,2,1,1,4,1,1,6,1,1,8,1,1,10,1,1,12,...] = [0;2,1,1,2p,1 ], p∈ℕ
1618
0.36787944117144232159552377016146086
0.69034 71261 14964 31946
Upper iterated exponential [ 144]
H
2
n
+
1
{\displaystyle {H}_{2n+1))
lim
n
→
∞
H
2
n
+
1
=
(
1
2
)
(
1
3
)
(
1
4
)
⋅
⋅
(
1
2
n
+
1
)
=
2
−
3
−
4
⋅
⋅
−
2
n
−
1
{\displaystyle \lim _{n\to \infty }{H}_{2n+1}=\textstyle \left({\frac {1}{2))\right)^{\left({\frac {1}{3))\right)^{\left({\frac {1}{4))\right)^{\cdot ^{\cdot ^{\left({\frac {1}{2n+1))\right)))))}={2}^{-3^{-4^{\cdot ^{\cdot ^{-2n-1))))))
2^-3^-4^-5^-6^ -7^-8^-9^-10^ -11^-12^-13 …
OEIS : A242760
[0;1,2,4,2,1,3,1,2,2,1,4,1,2,4,3,1,1,10,1,3,2,...]
0.69034712611496431946732843846418942
0.65836 55992 ...
Lower límit iterated exponential [ 145]
H
2
n
{\displaystyle {H}_{2n))
lim
n
→
∞
H
2
n
=
(
1
2
)
(
1
3
)
(
1
4
)
⋅
⋅
(
1
2
n
)
=
2
−
3
−
4
⋅
⋅
−
2
n
{\displaystyle \lim _{n\to \infty }{H}_{2n}=\textstyle \left({\frac {1}{2))\right)^{\left({\frac {1}{3))\right)^{\left({\frac {1}{4))\right)^{\cdot ^{\cdot ^{\left({\frac {1}{2n))\right)))))}={2}^{-3^{-4^{\cdot ^{\cdot ^{-2n))))))
2^-3^-4^-5^-6^ -7^-8^-9^-10^ -11^-12 …
[0;1,1,1,12,1,2,1,1,4,3,1,1,2,1,2,1,51,2,2,1,...]
0.6583655992...
3.14159 26535 89793 23846 [ Mw 119]
π number , Archimedes number [ 146]
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+\cdots +{\sqrt {2)))))))) _{n))
Sum[n=0 to ∞] {(-1)^n 4/(2n+1)}
T
OEIS : A000796
[3;7,15,1,292,1,1,1,2,1,3,1,14,2,1,1,2,2,2,...]
3.14159265358979323846264338327950288
0.46364 76090 00806 11621
Machin–Gregory series[ 147]
arctan
1
2
{\displaystyle \arctan {\frac {1}{2))}
∑
n
=
0
∞
(
−
1
)
n
x
2
n
+
1
2
n
+
1
=
1
2
−
1
3
⋅
2
3
+
1
5
⋅
2
5
−
1
7
⋅
2
7
+
⋯
For
x
=
1
/
2
{\displaystyle {\underset ((\text{For ))x=1/2\qquad \qquad }{\sum _{n=0}^{\infty }{\frac {(\!-1\!)^{n}\,x^{2n+1)){2n+1))={\frac {1}{2)){-}{\frac {1}{3\!\cdot \!2^{3))}{+}{\frac {1}{5\!\cdot \!2^{5))}{-}{\frac {1}{7\!\cdot \!2^{7))}{+}\cdots ))}
Sum[n=0 to ∞] {(-1)^n (1/2)^(2n+1) /(2n+1)}
A
OEIS : A073000
[0;2,6,2,1,1,1,6,1,2,1,1,2,10,1,2,1,2,1,1,1,...]
0.46364760900080611621425623146121440
1.90216 05831 04 [ Mw 120]
Brun 2 constant = Σ inverse of Twin primes [ 148]
B
2
{\displaystyle {B}_{\,2))
∑
(
1
p
+
1
p
+
2
)
p
,
p
+
2
:
prime
=
(
1
3
+
1
5
)
+
(
1
5
+
1
7
)
+
(
1
11
+
1
13
)
+
⋯
{\displaystyle \textstyle {\underset {p,\,p+2:{\text{ prime))}{\sum ({\frac {1}{p))+{\frac {1}{p+2)))))=({\frac {1}{3))\!+\!{\frac {1}{5)))+({\tfrac {1}{5))\!+\!{\tfrac {1}{7)))+({\tfrac {1}{11))\!+\!{\tfrac {1}{13)))+\cdots }
OEIS : A065421
[1; 1, 9, 4, 1, 1, 8, 3, 4, 4, 2, 2]
1.902160583104
0.87058 83799 75 [ Mw 121]
Brun 4 constant = Σ inv.prime quadruplets [ 149]
B
4
{\displaystyle {B}_{\,4))
∑
(
1
p
+
1
p
+
2
+
1
p
+
4
+
1
p
+
6
)
p
,
p
+
2
,
p
+
4
,
p
+
6
:
prime
{\displaystyle \textstyle {\sum ({\frac {1}{p))+{\frac {1}{p+2))+{\frac {1}{p+4))+{\frac {1}{p+6)))}\scriptstyle \quad {p,\;p+2,\;p+4,\;p+6:{\text{ prime))))
(
1
5
+
1
7
+
1
11
+
1
13
)
+
(
1
11
+
1
13
+
1
17
+
1
19
)
+
…
{\displaystyle \textstyle {\left({\tfrac {1}{5))+{\tfrac {1}{7))+{\tfrac {1}{11))+{\tfrac {1}{13))\right)}+\left({\tfrac {1}{11))+{\tfrac {1}{13))+{\tfrac {1}{17))+{\tfrac {1}{19))\right)+\dots }
OEIS : A213007
[0; 1, 6, 1, 2, 1, 2, 956, 3, 1, 1]
0.870588379975
0.63661 97723 67581 34307 [ Mw 122]
[ Ow 5]
Buffon constant[ 150]
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 }
Viète product
2/Pi
T
OEIS : A060294
[0;1,1,1,3,31,1,145,1,4,2,8,1,6,1,2,3,1,4,...]
1540 to 1603
0.63661977236758134307553505349005745
0.59634 73623 23194 07434 [ Mw 123]
Euler–Gompertz constant [ 151]
G
{\displaystyle {G))
∫
0
∞
e
−
n
1
+
n
d
n
=
∫
0
1
1
1
−
ln
n
d
n
=
1
1
+
1
1
+
1
1
+
2
1
+
2
1
+
3
1
+
3
/
⋯
{\displaystyle \!\int \limits _{0}^{\infty }\!\!{\frac {e^{-n)){1{+}n))\,dn=\!\!\int \limits _{0}^{1}\!\!{\frac {1}{1{-}\ln n))\,dn=\textstyle {\tfrac {1}{1+{\tfrac {1}{1+{\tfrac {1}{1+{\tfrac {2}{1+{\tfrac {2}{1+{\tfrac {3}{1+3{/\cdots ))))))))))))))
integral[0 to ∞] {(e^-n)/(1+n)}
OEIS : A073003
[0;1,1,2,10,1,1,4,2,2,13,2,4,1,32,4,8,1,1,1,...]
0.59634736232319407434107849936927937
i ··· [ Mw 124]
Imaginary number [ 152]
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
1501 to 1576
i
0.69777 46579 64007 98200 [ Mw 125]
Continued fraction constant, Bessel function [ 153]
C
C
F
{\displaystyle {C}_{CF))
I
1
(
2
)
I
0
(
2
)
=
∑
n
=
0
∞
n
n
!
n
!
∑
n
=
0
∞
1
n
!
n
!
=
1
1
+
1
2
+
1
3
+
1
4
+
1
5
+
1
6
+
1
/
⋯
{\displaystyle {\frac {I_{1}(2)}{I_{0}(2)))={\frac {\sum \limits _{n=0}^{\infty }{\frac {n}{n!n!))}{\sum \limits _{n=0}^{\infty }{\frac {1}{n!n!))))=\textstyle {\tfrac {1}{1+{\tfrac {1}{2+{\tfrac {1}{3+{\tfrac {1}{4+{\tfrac {1}{5+{\tfrac {1}{6+1{/\cdots ))))))))))))))
(Sum [n=0 to ∞] {n/(n!n!)}) / (Sum [n=0 to ∞] {1/(n!n!)})
OEIS : A052119
[0;1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,...] = [0;p+1 ], p∈ℕ
0.69777465796400798200679059255175260
2.74723 82749 32304 33305
Ramanujan nested radical [ 154]
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
A
[2;1,2,1,21,1,7,2,1,1,2,1,2,1,17,4,4,1,1,4,2,...]
2.74723827493230433305746518613420282
0.56714 32904 09783 87299 [ Mw 126]
Omega constant , Lambert W function [ 155]
Ω
{\displaystyle {\Omega ))
∑
n
=
1
∞
(
−
n
)
n
−
1
n
!
=
(
1
e
)
(
1
e
)
⋅
⋅
(
1
e
)
=
e
−
Ω
=
e
−
e
−
e
⋅
⋅
−
e
{\displaystyle \sum _{n=1}^{\infty }{\frac {(-n)^{n-1)){n!))=\,\left({\frac {1}{e))\right)^{\left({\frac {1}{e))\right)^{\cdot ^{\cdot ^{\left({\frac {1}{e))\right)))))=e^{-\Omega }=e^{-e^{-e^{\cdot ^{\cdot ^{-e))))))
Sum[n=1 to ∞] {(-n)^(n-1)/n!}
T
OEIS : A030178
[0;1,1,3,4,2,10,4,1,1,1,1,2,7,306,1,5,1,2,1,...]
0.56714329040978387299996866221035555
0.96894 61462 59369 38048
Beta (3) [ 156]
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))}{+}\cdots }
Sum[n=1 to ∞] {(-1)^(n+1) /(-1+2n)^3}
T
OEIS : A153071
[0;1,31,4,1,18,21,1,1,2,1,2,1,3,6,3,28,1,...]
0.96894614625936938048363484584691860
2.23606 79774 99789 69640
Square root of 5 , Gauss sum [ 157]
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 (n=5)\displaystyle \sum _{k=0}^{n-1}e^{\frac {2k^{2}\pi i}{n))=1+e^{\frac {2\pi i}{5))+e^{\frac {8\pi i}{5))+e^{\frac {18\pi i}{5))+e^{\frac {32\pi i}{5))}
Sum[k=0 to 4] {e^(2k^2 pi i/5)}
A
OEIS : A002163
[2;4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,...] = [2;4 ,...]
2.23606797749978969640917366873127624
3.35988 56662 43177 55317 [ Mw 127]
Prévost constant Reciprocal Fibonacci constant [ 158]
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 }
Fn : Fibonacci series
Sum[n=1 to ∞] {1/Fibonacci[n]}
I
OEIS : A079586
[3;2,1,3,1,1,13,2,3,3,2,1,1,6,3,2,4,362,...]
?
3.35988566624317755317201130291892717
2.68545 20010 65306 44530 [ Mw 128]
Khinchin's constant [ 159]
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))}
T
OEIS : A002210
[2;1,2,5,1,1,2,1,1,3,10,2,1,3,2,24,1,3,2,...]
1934
2.68545200106530644530971483548179569