Matemaattinen vakio – määritelmä, esimerkit ja merkitys

Nimi

Symboli

Arvo

Merkitys

Pi, Arkhimedeen vakio tai Ludofin luku.

π

≈3.141592653589793

Transsendenttiluku, joka on ympyrän kehän pituuden ja halkaisijan suhde. Se on myös yksikköympyrän pinta-ala.

E, Napierin vakio tai Eulerin luku (lausutaan "oilers").

e

≈2.718281828459045

Transsendentaaliluku, joka on luonnollisten logaritmien perusta, joskus myös "luonnollinen luku".

Kultainen suhde

φ

5 + 1 2 ≈ 1.618 {\displaystyle {\frac {{\sqrt {5}}+1}{2}}}\approx 1.618}

Se on suuremman arvon arvo jaettuna pienemmällä arvolla, jos se on yhtä suuri kuin arvojen summa jaettuna suuremmalla arvolla.

Neliöjuuri 2:sta, Pythagoraan vakio.

2 {\displaystyle {\sqrt {2}}}

≈ 1.414 {\displaystyle \approx 1.414}

Irrationaalinen luku, joka on sellaisen neliön lävistäjän pituus, jonka sivut ovat pituudeltaan 1. Tätä lukua ei voi kirjoittaa murtolukuna.

Arvo

Nimi

Symboli

LaTeX

Kaava

Tyyppi

OEIS

Jatkettu murto

3.24697960371746706105000976800847962

Hopea, Tutte-Beraha vakio

ς \displaystyle \varsigma }

2 + 2 cos ( 2 π / 7 ) = 2 + 2 + 7 + 7 7 + 7 7 + 7 7 + ⋯ 3 3 3 1 + 7 + 7 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)

T

A116425

[3;4,20,2,3,1,6,10,5,2,2,1,2,2,1,18,1,1,3,2,...]

1.09864196439415648573466891734359621

Pariisin vakio

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

Ramanujanin sisäkkäinen radikaali R₅

R 5 {\displaystyle R_{5}}

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+{\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

5:n neliöjuuri, Gaussin summa

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=0-4]{e^(2k^2 pi i/5)}

I

A002163

[2;4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,...]
= [2;(4),...]

3.62560990822190831193068515586767200

Gamma(1/4)

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 vakio, 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 - ∞]{(-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-Bouwkampin vakio

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 - ∞]{cos(pi/n)}

T

A085365

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

1.78107241799019798523650410310717954

Exp(gamma)
G-Barnesin funktio

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 {\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 - ∞]{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-Kinkelinin vakio

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

Schwarzschildin kartiovakio

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 - ∞]{2^n/n!}

T

A072334

[7;2,1,1,3,18,5,1,1,6,30,8,1,1,9,42,11,1,...]
= [7,2,(1,1,n,4*n+6,n+2)], n = 3, 6, 9 jne.

1.01494160640965362502120255427452028

Giesekingin vakio

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 vakio

{\displaystyle {\displaystyle {\varpi }}

π G = 4 2 π ( 1 4 ! ) 2 {\displaystyle \pi \,{G}=4{\sqrt {\tfrac {\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

Gaussin vakio

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 k - g e o m e t r i k m e a n {\displaystyle {\underset {Agm:\;Aritmeettis-geometrinen\;keskiarvo}{{\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 - ∞] {1/(1-kymmenesluku(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 - ∞} (1-1/kymmenesluku(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

Neliön ja ympyrän tai ympyrän sisäänkirjoitetun ympyrän suhde.

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}{1}}}+{\frac {1}{3}}-{\frac {1}{5}}-{\frac {1}{7}}+{\frac {1}{9}}+{\frac {1}{11}}}-\dots }

sum[n=1 - ∞]{(-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-vakio

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}}\,dx}

N[int[0 - ∞] {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

Luvun e neliöjuuri

e {\displaystyle {\sqrt {e}}}

∑ n = 0 ∞ 1 2 n 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}{1}{1}}+{\frac {1}{2}}}+{\frac {1}{8}}}+{\frac {1}{48}}}+\cdots }

sum[n=0 - ∞]{1/(2^n n!)}

T

A019774

[1;1,1,1,1,5,1,1,1,9,1,1,1,13,1,1,1,17,1,1,1,21,1,1,1,...]
= [1;1,(1,1,1,4p+1)], p∈ℕ

i

Kuvitteellinen luku

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-Ramanujanin vakio

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 vakio

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 {{\frac {4}{1}}{-{{\frac {4}{3}}{+}{\frac {4}{5}}{-}{\frac {4}{7}}{+}{\frac {4}{9}}-\dots }{{\frac {1}{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

Hyperbolinen tangentti (1)

t h 1 {\displaystyle th\,1}

e - 1 e 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;(2p+1)], p∈ℕ

0.69777465796400798200679059255175260

Jatkuva Fraktio vakio

C C F {\displaystyle {C}_{CF}}

J 1 ( 2 ) J 0 ( 2 ) T u o n k i 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}{1}}}+{\frac {1}{1}{1}{1}}}+{\frac {1}{4}}}+{\frac {1}{36}}}+{\frac {1}{576}}}+{\dots }}}}

(summa {n=0 - inf} n/(n!n!))) /(summa {n=0 - inf} 1/(n!n!))

A052119

[0;1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,...]
= [0;(p+1)], p∈ℕ

0.36787944117144232159552377016146086

Napierin käänteisvakio

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}{1!}}+{\frac {1}{2!}}-{\frac {1}{3!}}+{\frac {1}{4!}}-{\frac {1}{5!}}+\dots }

sum[n=2 - ∞]{(-1)^n/n!}

T

A068985

[0;2,1,1,2,1,1,4,1,1,1,6,1,1,1,8,1,1,1,10,1,1,1,12,...]
= [0;2,1,(1,2p,1)], p∈ℕ

2.71828182845904523536028747135266250

Napierin vakio

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}{1}}}+{\frac {1}{2!}}+{\frac {1}{3!}}+{\frac {1}{4!}}+{\frac {1}{5!}}+{\cdots}

Sum[n=0-∞]{1/n!}

T

A001113

[2;1,2,1,1,4,1,1,1,6,1,1,1,8,1,1,1,10,1,1,1,12,1,...]
= [2;(1,2p,1)], p∈ℕ

0.49801566811835604271369111746219809
- 0.15494982830181068512495513048388 i

i:n kertolasku

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,...]
- [0;2,125,2,18,1,2,1,1,19,1,1,1,2,3,34,...] i

0.43828293672703211162697516355126482
+ 0.36059247187138548595294052690600 i

Infinite
i:n tetraatio

∞ 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^{i^{\cdot ^{\cdot ^{i}}}}} _{n}}

i^i^i^i^...

C

A077589
A077590

[0;2,3,1,1,4,2,2,1,10,2,1,3,1,8,2,1,2,1, ...]
+ [0;2,1,3,2,2,3,1,5,5,1,2,1,10,10,6,1,1...] i

0.56755516330695782538461314419245334

Infinite-moduuli
i:n tetraatio

∞ i | ∞ i | ∞ i | ∞ i | ∞ i | ∞ i | ∞ i | ∞ i | ∞ i | ∞ i | ∞ i | ∞ i | ∞ i | ∞ i|}

lim n → ∞ | n i | = | lim n → ∞ i 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^i^...).

A212479

[0;1,1,3,4,1,58,12,1,51,1,4,12,1,1,2,2,3,...]

0.26149721284764278375542683860869585

Meissel-Mertensin vakio

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)} ..... p: alkuluvut

A077761

[0;3,1,4,1,2,5,2,1,1,1,1,13,4,2,4,2,1,33,...]

1.9287800...

Wrightin vakio

ω \displaystyle \omega }

⌊ 2 2 2 ⋅ ⋅ 2 ω ⌋ {\displaystyle \left\lfloor 2^{2^{2^{\cdot ^{\cdot ^{2^{\omega }}}}}}\right\rfloor } = primos: {\displaystyle \quad } ⌊ 2 ω ⌋ {\displaystyle \left\lfloor 2^{\omega }\right\rfloor } =3, ⌊ 2 2 ω ⌋ {\displaystyle \left\lfloor 2^{2^{\omega }}\right\rfloor } =13, ⌊ 2 2 2 ω ⌋ {\displaystyle \left\lfloor 2^{2^{2^{2^{\omega }}}\right\rfloor } =16381, ... {\displaystyle \dots }

A086238

[1; 1, 13, 24, 2, 1, 1, 3, 1, 1, 3]

0.37395581361920228805472805434641641

Artinin vakio

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)} ...... p_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

Feigenbaumin vakio 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

Feigenbaumin vakio 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

Kuusikulmainen 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)} β ( 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 - ∞]{(-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-vakio₂ = Σ käänteinen kaksoispriimi

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-vakio₄ = Σ käänteisluku kaksoisprimasta

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, Arkhimedeen vakio

π \displaystyle \pi }

lim n → ∞ 2 n 2 - 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 - ∞]{(-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}} ... Tetraation alaraja

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 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

Bernsteinin vakio

β 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 ja Richmond

Q {\displaystyle Q}

∏ n = 1 ∞ ( 1 - 1 2 n ) = ( 1 - 1 2 1 ) ( 1 - 1 2 2 ) ( 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 - ∞]{1-1/2^n}

A048651

0.31830988618379067153776752674502872

Piin käänteisluku, 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

Weierstraßin vakio

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-vakio

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 - ∞]{(-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

Eulerin luku

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 - ∞]|sum[k=0 - ∞]{((-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-sarja

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 2 ⋯ {\displaystyle {\frac {\sqrt {2}{2}{2}}\cdot {\frac {\sqrt {2+{\sqrt {2}}}}{2}}}\cdot {\frac {\sqrt {2+{\sqrt {2+{\sqrt {2+{\sqrt {2}}}}{2}}}\cdot }

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 -vakio

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 - ∞]{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 Rajavakio

λ λ \displaystyle \lambda }

A033259

[0;1,1,1,27,1,1,1,8,2,154,2,4,1,5,...]

0.69314718055994530941723212145817657

Logaritmi 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 - ∞]{(-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 Dream₁ J.Bernoulli

I 1 {\displaystyle I_{1}}

∑ n = 1 ∞ ( - 1 ) n + 1 n 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)} 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 - ∞]{(-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

Matkustava myyntimies Nielsen-Ramanujan 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}}}}-\pisteet }

Sum[n=1 - ∞]{((-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

Katalaani vakio

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 - ∞]{(-1)^n/(2n+1)^2}

I

A006752

[0;1,10,1,8,1,88,4,1,1,7,22,1,2,...]

1.05946309435929526456182529494634170

Puoliäänten välisen etäisyyden suhdeluku

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 - ∞]{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 - ∞]{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

Favardin vakio

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 - ∞]{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

Kuutiojuuri 2:sta, 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 Dream₂ 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

Muovinen numero

ρ \displaystyle \rho }

1 + 1 + 1 + 1 + 1 + 1 + ⋯ 3 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

Neliöjuuri 2:sta, Pythagoraan vakio.

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}}}{oikea)\left(1{-}{\frac {1}{3}}}\oikea)\left(1{+}{\frac {1}{5}}\oikea)...}

prod[n=1 - ∞]{1+(-1)^(n+1)/(2n-1)}

I

A002193

[1;2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,...]
= [1;(2),...]

1.44466786100976613365833910859643022

Steinerin luku

e 1 e {\displaystyle e^{\frac {1}{e}}}}

e 1 / e {\displaystyle e^{1/e}} ... Tetraation yläraja

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

Wallisin tuote

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 }\cdots }

T

A019669

[1;1,1,3,31,1,145,1,4,2,8,1,6,1,2,3,1...]

1.60669515241529176378330152319092458

Erdős-Borweinin vakio

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}{1}}}+{\frac {1}{3}}}+{\frac {1}{7}}+{\frac {1}{15}}+\cdots \,\!}

sum[n=1 - ∞]{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, kultainen leikkaus

φ \displaystyle \varphi }

1 + 5 2 = 1 + 1 + 1 + 1 + 1 + ⋯ {\displaystyle {\frac {1+{\sqrt {5}}{2}}={\sqrt {1+{\sqrt {1+{\sqrt {1+{\sqrt {1+{\sqrt {1+\cdots }}}}}}}}}

(1+5^(1/2))/2

I

A001622

[0;1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,...]
= [0;(1),...]

1.64493406684822643647241516664602519

Zeta(2)

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}}=\summa _{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 - ∞]{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

Somosin kvadraattinen toistuvuusvakio

σ \displaystyle \sigma }

1 2 3 ⋯ = 1 1 / 2 ; 2 1 / 4 ; 3 1 / 8 ⋯ {\displaystyle {\sqrt {1{\sqrt {1{\sqrt {2{\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

Theodoruksen vakio

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;(1,2),...]

1.75793275661800453270881963821813852

Kasnerin numero

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-Levinin vakio

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

Universaali parabolinen vakio

P 2 {\displaystyle P_{\,2}}

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

Pronssi numero

σ R r {\displaystyle \sigma _{\,Rr}}

3 + 13 2 = 1 + 3 + 3 + 3 + 3 + 3 + ⋯ {\displaystyle {\frac {3+{\sqrt {13}}{2}}=1+{\sqrt {3+{\sqrt {3+{\sqrt {3+{\sqrt {3+{\sqrt {3+{\sqrt {3+\cdots }}}}}}}}}

(3+sqrt 13)/2

I

A098316

[3;3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,...]
= [3;(3),...]

2.37313822083125090564344595189447424

Lévy-vakio₂

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

2 pi:n neliöjuuri

2 π {\displaystyle {\sqrt {2\pi}}

2 π = lim n → ∞ n ! e n 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-Schneiderin vakio

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 vakio

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 - ∞]{(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-vakio

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

Fibonacci-vakio reciprocal

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}{1}}+{\frac {1}{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

Juuri 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

Frodan vakio

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 neliö

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 - ∞]{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

Gelfondin vakio

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 - ∞]{(pi^n)/n!}

T

A039661

[23;7,9,3,1,1,591,2,9,1,2,34,1,16,1,30,1,...]