Constantă matematică | număr, care are o semnificație specială pentru calcule

Nume

Simbol

Valoare

Adică

Pi, constanta lui Arhimede sau numărul lui Ludoph

π

≈3.141592653589793

Un număr transcendental care reprezintă raportul dintre lungimea circumferinței unui cerc și diametrul acestuia. Este, de asemenea, aria cercului unitar.

E, constanta lui Napier sau numărul lui Euler (pronunțat "oilers")

e

≈2.718281828459045

Un număr transcendental care reprezintă baza logaritmilor naturali, numit uneori "număr natural".

Raportul de aur

φ

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

Este valoarea unei valori mai mari împărțită la o valoare mai mică, dacă aceasta este egală cu valoarea sumei valorilor împărțită la valoarea mai mare.

Rădăcina pătrată a lui 2, constanta lui Pitagora

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

≈ 1.414 {\displaystyle \aprox. 1.414}

Un număr irațional care reprezintă lungimea diagonalei unui pătrat cu laturile de lungime 1. Acest număr nu poate fi scris sub formă de fracție.

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3.24697960371746706105000976800847962

Argint, Tutte-Beraha constant

ς {\displaystyle \varsigma }

2 + 2 cos ( 2 π / 7 ) = 2 + 2 + 2 + 7 + 7 7 + 7 7 + 7 7 + ⋯ 3 3 3 1 + 7 + 7 + 7 7 + 7 7 + ⋯ 3 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+7{\sqrt[{3}]{\,7+\cdots }}}}}}}{1+{\sqrt[{3}]{7+7{\sqrt[{3}]{7+7{\sqrt[{3}]{\,7+\cdots }}}}}}}}}

2+2 cos(2Pi/7)

T

A116425

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

1.09864196439415648573466891734359621

Paris constant

C P a {\displaystyle C_{Pa}}

∏ n = 2 ∞ 2 φ φ + φ n , φ = F i {\displaystyle \prod _{n=2}^{\infty }{\frac {2\varphi }{\varphi +\varphi _{n}}}\;,\varphi ={Fi}}}

I

A105415

[1;10,7,3,1,3,1,5,1,4,2,7,1,2,3,22,1,2,5,2,1,...]

2.74723827493230433305746518613420282

Ramanujan radical imbricate R₅

R 5 {\displaystyle R_{5}}

5 + 5 + 5 + 5 - 5 + 5 + 5 + 5 + 5 - ⋯ = 2 + 5 + 5 + 15 - 6 5 5 2 {\displaystyle \scriptstyle {\sqrt {5+{\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

Rădăcina pătrată a lui 5, suma Gauss

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 la 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 constant, Marvin Ray Burns

C M R B {\displaystyle C_{_{MRB}}}

∑ n = 1 ∞ ( - 1 ) n ( n 1 / n - 1 ) = - 1 1 1 + 2 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 la ∞]{(-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

Constanta Kepler-Bouwkamp

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}}\dreapta)=\cos \left({\frac {\pi }{3}}\dreapta)\cos \left({\frac {\pi }{4}}\dreapta)\cos \left({\frac {\pi }{5}}\dreapta)\dots }

prod[n=3 la ∞]{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)
Funcția G-Barnes

e γ {\displaystyle e^{\gamma }}}

∏ n = 1 ∞ e 1 n 1 + 1 n = ∏ n = 0 ∞ ( ∏ k = 0 ( 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 \alege k}}\right)^{\frac {1}{n+1}}}=}}

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

Prod[n=1 până la ∞]{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

Constanta Glaisher-Kinkelin

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 \sum \limits _{k=0}^{n}\left(-1\dreapta)^{k}{{\binom {n}{k}}}\left(k+1\dreapta)^{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

Constanta conică Schwarzschild

e 2 {\displaystyle e^{2}}

∑ n = 0 ∞ 2 n 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 la ∞]{2^n/n!}

T

A072334

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

1.01494160640965362502120255427452028

Constanta Gieseking

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 constantă

ϖ {\displaystyle {\varpi }}

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

4 sqrt(2/pi) (1/4!)^2

T

A062539

[2;1,1,1,1,1,4,1,2,5,1,1,1,14,9,2,6,2,9,4,1,...]

0.83462684167407318628142973279904680

Constanta Gauss

G {\displaystyle {G}}

1 a g m ( 1 , 2 ) = 4 2 ( 1 4 ! ) 2 π 3 / 2 A g m : A r i t h m e t i c - g e o m e t r i c m e a n {\displaystyle {\underset {Agm:\;Arithmetic-geometric\;mean}{{{\frac {1}{\mathrm {agm} (1,{\sqrt {2}})}}={\frac {4{\sqrt {2}}}\,({\tfrac {1}{4}}!)^{2}}}{\pi ^{3/2}}}}}}

(4 sqrt(2)(1/4!)^2)/pi^(3/2)

T

A014549

[0;1,5,21,3,4,14,1,1,1,1,1,3,1,15,1,3,7,1,...]

1.01734306198444913971451792979092052

Zeta(6)

ζ ( 6 ) {\displaystyle \zeta (6)}

π 6 945 = ∏ n = 1 ∞ 1 1 1 - p n - 6 p n : p r i m o = 1 1 1 - 2 - 6 ⋅ 1 1 1 - 3 - 6 ⋅ 1 1 1 - 5 - 6 . . . . {\displaystyle {\frac {\pi ^{6}}}{945}}=\prod _{n=1}^{\infty }{\underset {p_{n}:\,{primo}}{\frac {1}{{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 la ∞] {1/(1-ithprime(n)^-6)}

T

A013664

[1;57,1,1,1,15,1,6,3,61,1,5,3,1,6,1,3,3,6,1,...]

0,60792710185402662866327677925836583

Constante de Hafner-Sarnak-McCurley

1 ζ ( 2 ) {\displaystyle {\frac {1}{\zeta (2)}}}}

6 π 2 = ∏ n = 0 ∞ ( 1 - 1 p n 2 ) p n : p r i m o = ( 1 - 1 2 2 ) ( 1 - 1 3 2 ) ( 1 - 1 5 2 ) ... {\displaystyle {\frac {\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 până la ∞} (1-1/ithprime(n)^2)

T

A059956

[0;1,1,1,1,4,2,4,7,1,4,2,3,4,10,1,2,1,1,1,...]

1.11072073453959156175397024751517342

Raportul dintre un pătrat și cercurile circumscrise sau înscrise

π 2 2 2 {\displaystyle {\frac {\pi }{2{\sqrt {2}}}}}

∑ n = 1 ∞ ( - 1 ) ⌊ n - 1 2 ⌋ 2 n + 1 = 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}{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 la ∞]{(-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

Constanta Fransén-Robinson

F {\displaystyle {F}}

∫ 0 ∞ 1 Γ ( x ) d x . = e + ∫ 0 ∞ e - x π 2 + ln 2 x d x {\displaystyle \int _{0}^{\infty }{\frac {1}{\Gamma (x)}}\,dx.=e+\int _{0}^{\infty }{\frac {e^{-x}}}{\pi ^{2}+\ln ^{2}x}}}\,dx}

N[int[0 la ∞] {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

Rădăcina pătrată a numărului e

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

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

sum[n=0 la ∞]{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,(1,1,4p+1)], p∈ℕ

i

Număr imaginar

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

Constanta Hermite-Ramanujan

R {\displaystyle {R}}

e π 163 {\displaystyle e^{\pi {\sqrt {163}}}}

e^(π sqrt(163))

T

A060295

[262537412640768743;1,1333462407511,1,8,1,1,5,...]

4.81047738096535165547303566670383313

John constant

γ {\displaystyle \gamma }

i i i = i - i = i 1 i = ( 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 {\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}}-puncte }{{\frac {1}{1}{1}}{-}{\frac {1}{2}}{+}{\frac {1}{3}}{-}{\frac {1}{4}}{+}{\frac {1}{5}}}-puncte }}}

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

Tangentă hiperbolică (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;1,3,3,5,7,9,11,13,15,17,19,21,23,25,27,...]
= [0;(2p+1)], p∈ℕ

0.69777465796400798200679059255175260

Constantă de fracție continuă

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

J 1 ( 2 ) J 0 ( 2 ) F u n c ț i o n e J k ( ) B e s s e l = ∑ n = 0 ∞ n n ! n ! n ! ∑ n = 0 ∞ 1 n ! 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 {Funcție}{\frac {J_{1}(2)}{J_{0}(2)}}}}={\frac {\sum \limits _{n=0}^{\infty }{\frac {n}{n!n!}}}{\sum \sum \limits _{n=0}^{\infty }{\frac {1}{n!n!}}}}={\frac {{\frac {0}{1}}+{\frac {1}{1}{1}}+{\frac {2}{4}}+{\frac {3}{36}}+{\frac {4}{576}}+\dots }{{\frac {1}{1}{1}}}+{\frac {1}{1}{1}}}+{\frac {1}{4}}+{\frac {1}{36}}+{\frac {1}{576}}+\dots }}}

(suma {n=0 la inf} n/(n!n!)) /(suma {n=0 până la inf} 1/(n!n!))

A052119

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

0.36787944117144232159552377016146086

Constanta inversă a lui Napier

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

∑ n = 0 ∞ ( - 1 ) n n n ! = 1 0 ! - 1 1 ! + 1 2 ! - 1 3 ! + 1 4 ! - 1 5 ! + ... {\displaystyle \sum _{n=0}^{\infty }{\frac {(-1)^{n}}}{n!}}={\frac {1}{0!}}-{\frac {1}{1!}}+{\frac {1}{2!}}-{\frac {1}{3!}}+{\frac {1}{4!}}-{\frac {1}{5!}}+\dots }

sum[n=2 până la ∞]{(-1)^n/n!}

T

A068985

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

2.71828182845904523536028747135266250

Napier constant

e {\displaystyle e}

∑ n = 0 ∞ 1 n ! = 1 0 ! + 1 1 + 1 2 ! + 1 3 ! + 1 4 ! + 1 5 ! + ⋯ {\displaystyle \sum _{n=0}^{\infty }{\frac {1}{n!}}={\frac {1}{0!}}+{\frac {1}{1}{1}}+{\frac {1}{2!}}+{\frac {1}{3!}}+{\frac {1}{4!}}+{\frac {1}{5!}}+\cdots }

Suma[n=0 până la ∞]{1/n!}

T

A001113

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

0.49801566811835604271369111746219809
- 0.15494982830181068512495513048388 i

Factorial de i

i ! {\displaystyle i\,!}

Γ ( 1 + i ) = i Γ ( i ) {\displaystyle \Gamma (1+i)=i\,\Gamma (i)}

Gamma(1+i)

C

A212877 A212878

[0;6,2,4,1,8,1,46,2,2,3,5,1,10,7,5,1,7,2,...]
- [0;2,125,2,18,1,2,1,1,19,1,1,1,2,3,34,...] i

0.43828293672703211162697516355126482
+ 0.36059247187138548595294052690600 i

Infinit
Tetrație de i

∞ i {\displaystyle {}^{\infty }i}

lim n → ∞ n i = lim n → ∞ i i i ⋅ ⋅ i ⏟ n {\displaystyle \lim _{n\to \infty }{}^{n}i=\lim _{n\to \infty }\underbrace {i^{i^{\cdot ^{\cdot ^{i}}}}} _{n}}

i^i^i^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

Modulul de
Infinit
Tetrația lui i

| ∞ i | {\displaystyle |{}^{\infty }i|}

lim n → ∞ | n i | n i | = | lim n → ∞ i i i ⋅ ⋅ i i ⏟ n | {\displaystyle \lim _{n\\to \infty }\left|{}^{n}i\right|=\left|\lim _{n\to \infty }\underbrace {i^{i^{i^{\cdot ^{\cdot ^{i}}}}} {\n} {\n}Dreapta|}

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

Constanta Meissel-Mertens

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: prime

A077761

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

1.9287800...

Constanta Wright

ω {\displaystyle \omega }

⌊ 2 2 2 2 2 ⋅ ⋅ 2 ω ⌋ {\displaystyle \left\lfloor 2^{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 2 ω ⌋ {\displaystyle \left\lfloor 2^{2^{2^{\omega }}}\right\rfloor } =16381, ... {\displaystyle \dots }

A086238

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

0.37395581361920228805472805434641641

Constanta Artin

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

Constanta Feigenbaum 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

Constanta Feigenbaum α

α {\displaystyle \alpha }

lim n → ∞ d n d n d n + 1 {\displaystyle \lim _{n\to \infty }{\frac {d_{n}}}{d_{n+1}}}}

T

A006891

[2;1,1,85,2,8,1,10,16,3,8,9,2,1,40,1,2,3,...]

5.97798681217834912266905331933922774

Hexagonal Madelung Constant ₂

H 2 ( 2 ) {\displaystyle H_{2}(2)}

π ln ( 3 ) 3 {\displaystyle \pi \ln(3){\sqrt {3}}}}}

Pi Log[3]Sqrt[3]

T

A086055

[5;1,44,2,2,1,15,1,1,12,1,65,11,1,3,1,1,...]

0.96894614625936938048363484584691860

Beta(3)

β ( 3 ) {\displaystyle \beta (3)}

π 3 32 = ∑ n = 1 ∞ - 1 n + 1 ( - 1 + 2 n ) 3 = 1 1 3 - 1 3 3 + 1 5 3 - 1 7 3 + ... {\displaystyle {\frac {\frac {\pi ^{3}}{32}}=\sum _{n=1}^{\infty }{\frac {-1^{n+1}}{{(-1+2n)^{3}}}={\frac {1}{1}{1^{3}}}{-}{\frac {1}{3^{3}}}{+}{\frac {1}{5^{3}}}{-}{}{\frac {1}{7^{3}}}{+}\dots }

Sum[n=1 la ∞]{(-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 constant₂ = Σ prime gemene inverse

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

Constanta Brun₄ = Σ inversa primului geamăn

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, constanta lui Arhimede

π {\displaystyle \pi }

lim n → ∞ 2 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 până la ∞]{(-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}} ... Limita inferioară a tetrației

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

Constanta Bernstein

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 și 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}}}\dreapta)=\left(1{-}{\frac {1}{2^{1}}}\dreapta)\left(1{-}{\frac {1}{2^{2}}}\dreapta)\left(1{-}{\frac {1}{2^{3}}}\dreapta)\dots }

prod[n=1 la ∞]{1-1/2^n}

A048651

0.31830988618379067153776752674502872

Inversa lui Pi, Ramanujan

1 π {\displaystyle {\frac {1}{\pi }}}}

2 2 2 9801 ∑ n = 0 ∞ ( 4 n ) ! ( 1103 + 26390 n ) ( n ! ) 4 396 4 n {\displaystyle {\frac {\frac {2{\sqrt {2}}}{9801}}}\sum _{n=0}^{\infty }{\frac {(4n)!(1103+26390n)}{(n!)^{4}396^{4n}}}}

T

A049541

[0;3,7,15,292,1,1,1,2,1,3,1,14,2,1,1,...]

0.47494937998792065033250463632798297

Weierstraß constant

W W E {\displaystyle W_{_{WE}}

e π 8 π 4 ∗ 2 3 / 4 ( 1 4 ! ) 2 {\displaystyle {\frac {e^{\frac {\pi }{8}}}{\sqrt {\pi }}}{4*2^{3/4}{{({\frac {1}{4}}!)^{2}}}}}

(E^(Pi/8) Sqrt[Pi])/(4 2^(3/4) (1/4)!^2)

T

A094692

[0;2,9,2,11,1,6,1,4,6,3,19,9,217,1,2,...]

0.56714329040978387299996866221035555

Constanta Omega

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 la ∞]{(-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

Numărul lui Euler

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 până la ∞]|sum[k=0 până la ∞]{((-1)^k)/(2^n+k)}

?

A001620

[0;1,1,2,1,2,1,4,3,13,5,1,1,8,1,2,...]

0.60459978807807261686469275254738524

Seria Dirichlet

π 3 3 3 {\displaystyle {\frac {\pi }{3{\sqrt {3}}}}}

∑ n = 1 ∞ 1 n ( 2 n n n ) = 1 - 1 2 + 1 4 - 1 5 + 1 7 - 1 8 + ⋯ {\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n{2n \alege 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 + 2 2 ⋯ {\displaystyle {\frac {\sqrt {2}}{2}}\cdot {\frac {\sqrt {2+{\sqrt {2}}}}{2}}\cdot {\frac {\sqrt {2+{\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}{2}}\cdots }

T

A060294

[0;1,1,1,3,31,1,145,1,4,2,8,1,6,1,2,3,1,4,...]

0.66016181584686957392781211001455577

Constanta primelor gemene

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 la ∞]{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

Constanta limită Laplace

λ {\displaystyle \lambda }

A033259

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

0.69314718055994530941723212145817657

Logaritm de 2

L n ( 2 ) {\displaystyle Ln(2)}

∑ n = 1 ∞ ( - 1 ) n + 1 n = 1 1 1 - 1 2 + 1 3 - 1 4 + 1 5 - ⋯ {\displaystyle \sum _{n=1}^{\infty }{\frac {(- - -1)^{n+1}}}{n}}={{\frac {1}{1}{1}}}-{\frac {1}{2}}+{\frac {1}{3}}-{\frac {1}{4}}+{\frac {1}{5}}-\cdots }

Sum[n=1 la ∞]{(-1)^(n+1)/n}

T

A002162

[0;1,2,3,1,6,3,1,1,2,1,1,1,1,3,10,...]

0.78343051071213440705926438652697546

Visul celui de-al doilea an de liceu₁ J.Bernoulli

I 1 {\displaystyle I_{1}}

∑ n = 1 ∞ ( - 1 ) n + 1 n n n = 1 - 1 2 2 2 + 1 3 3 - 1 4 4 + 1 5 5 + ... {\displaystyle \sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n^{n}}}=1-{\frac {1}{2^{2}}}+{\frac {1}{3^{3}}}-{\frac {1}{4^{4}}}+{\frac {1}{5^{5}}}+\dots }

Sum[ -(-1)^n /n^n]

T

A083648

[0;1,3,1,1,1,1,1,1,2,4,7,2,1,2,1,1,1,...]

0.78539816339744830961566084581987572

Dirichlet beta(1)

β ( 1 ) {\displaystyle \beta (1)}

π 4 = ∑ n = 0 ∞ ( - 1 ) n 2 n + 1 = 1 1 1 - 1 3 + 1 5 - 1 7 + 1 9 - ⋯ {\displaystyle {\frac {\pi }{4}}=\sum _{n=0}^{\infty }{\frac {(- -1)^{n}}}{2n+1}}={{\frac {1}{1}{1}}}-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+{\frac {1}{9}}-\cdots }

Sum[n=0 până la ∞]{(-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

Vânzătorul ambulant Nielsen-Ramanujan

ζ ( 2 ) 2 {\displaystyle {\frac {\zeta (2)}{2}}}}

π 2 12 = ∑ n = 1 ∞ ( - 1 ) n + 1 n 2 = 1 1 1 2 - 1 2 2 + 1 3 2 - 1 4 2 + 1 5 2 - ... {\displaystyle {\frac {\frac {\pi ^{2}}{12}}=\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}}{n^{2}}}={\frac {1}{1}{1^{2}}}{-}{\frac {1}{2^{2}}}{+}{\frac {1}{3^{2}}}{-}{\frac {1}{4^{2}}}{+}{\frac {1}{5^{2}}}}-puncte }

Sum[n=1 la ∞]{((-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

Constanta catalană

C {\displaystyle C}

∑ n = 0 ∞ ( - 1 ) n ( 2 n + 1 ) 2 = 1 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 la ∞]{(-1)^n/(2n+1)^2}

I

A006752

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

1.05946309435929526456182529494634170

Raportul dintre distanța dintre semitonuri

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 1 4 + 1 2 4 + 1 3 4 + 1 4 4 + 1 5 4 + ... {\displaystyle {\frac {\frac {\pi ^{4}}{90}}=\sum _{n=1}^{\infty }{\frac {1}{n^{4}}}={\frac {1}{1}{1^{4}}}+{\frac {1}{2^{4}}}+{\frac {1}{3^{4}}}+{\frac {1}{4^{4}}}+{\frac {1}{5^{4}}}+\dots }

Sum[n=1 la ∞]{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 | 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 1 3 + 1 2 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}{1^{3}}}+{\frac {1}{2^{3}}}+{\frac {1}{3^{3}}}+{\frac {1}{4^{3}}}+{\frac {1}{5^{3}}}+{\cdots \,\!}

Sum[n=1 la ∞]{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

Constanta Favard

3 4 ζ ( 2 ) {\displaystyle {\tfrac {3}{4}}}\zeta (2)}

π 2 8 = ∑ n = 0 ∞ 1 ( 2 n - 1 ) 2 = 1 1 1 2 + 1 3 2 + 1 5 2 + 1 7 2 + ... {\displaystyle {\frac {\frac {\pi ^{2}}{8}}=\sum _{n=0}^{\infty }{\frac {1}{(2n-1)^{2}}}={\frac {1}{1}{1^{2}}}+{\frac {1}{3^{2}}}+{\frac {1}{5^{2}}}+{\frac {1}{7^{2}}}+\dots }

sum[n=1 la ∞]{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

Rădăcină cubică de 2, constantă 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

Visul celui de-al doilea an de liceu₂ 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

Număr de plastic

ρ {\displaystyle \rho }

1 + 1 + 1 + 1 + 1 + ⋯ 3 3 3 3 3 {\displaystyle {\sqrt[{3}]{1+{\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

Rădăcina pătrată a lui 2, constanta lui Pitagora

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

∏ n = 1 ∞ 1 + ( - 1 ) n + 1 2 n - 1 = ( 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}{1}}\dreapta)\left(1{-}{\frac {1}{3}}\dreapta)\left(1{+}{\frac {1}{5}}\dreapta)...}

prod[n=1 la ∞]{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

Numărul Steiner

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

e 1 / e {\displaystyle e^{1/e}} ... Limita superioară a tetrației

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

Produsul Wallis

π / 2 {\displaystyle \pi /2}

∏ n = 1 ∞ ( 4 n 2 4 n 2 4 n 2 - 1 ) = 2 1 ⋅ 2 3 ⋅ 4 3 ⋅ 4 5 ⋅ 6 5 ⋅ 6 7 ⋅ 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}}\cdot {\frac {8}{9}}\cdots }

T

A019669

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

1.60669515241529176378330152319092458

Constanta Erdős-Borwein

E B {\displaystyle E_{\,B}}

∑ n = 1 ∞ 1 2 n - 1 = 1 1 1 + 1 3 + 1 3 + 1 7 + 1 15 + ⋯ {\displaystyle \sum _{n=1}^{\infty }{\frac {1}{2}{2^{n}-1}}}={\frac {1}{1}{1}}}+{\frac {1}{3}}+{\frac {1}{7}}+{\frac {1}{15}}+\cdots \,\!}

sum[n=1 la ∞]{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, raportul de aur

φ {\displaystyle \varphi }

1 + 5 2 = 1 + 1 + 1 + 1 + 1 + 1 + ⋯ {\displaystyle {\frac {1+{\sqrt {5}}}{2}}}={\sqrt {1+{\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 1 2 + 1 2 2 + 1 3 2 + 1 4 2 + ⋯ {\displaystyle {\frac {\pi ^{2}}{6}}}=\suma _{{n=1}^{\infty }{\frac {1}{n^{2}}}={\frac {1}{1}{1^{2}}}+{\frac {1}{2^{2}}}+{\frac {1}{3^{2}}}+{\frac {1}{4^{2}}}+\cdots }

Sum[n=1 la ∞]{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

Constanta de recurență pătratică a lui Somos

σ {\displaystyle \sigma }

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

T

A065481

[1;1,1,1,21,1,1,1,6,4,2,1,1,2,1,3,1,13,13,...]

1.73205080756887729352744634150587237

Constanta Theodorus

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

Numărul Kasner

R {\displaystyle R}

1 + 2 + 3 + 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

Constanta Carlson-Levin

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

Constanta parabolică universală

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

ln ( 1 + 2 ) + 2 {\displaystyle \lnn(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

Număr de bronz

σ R r r {\displaystyle \sigma _{\,Rr}}

3 + 13 2 = 1 + 3 + 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

Constanta Lévy₂

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

rădăcina pătrată a 2 pi

2 π {\displaystyle {\sqrt {2\pi }}}

2 π = lim n → ∞ n ! e n 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

Constanta Gelfond-Schneider

G G G S {\displaystyle G_{_{{\\,GS}}}

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

2^sqrt{2}

T

A007507

[2;1,1,1,72,3,4,1,3,2,1,1,1,14,1,2,1,1,3,1,...]

2.68545200106530644530971483548179569

Khintchin constant

K 0 {\displaystyle K_{\,0}}

∏ n = 1 ∞ [ 1 + 1 n ( n + 2 ) ] ln n / ln 2 {\displaystyle \prod _{n=1}^{\infty }\left[{1+{1 \1 peste n(n+2)}}}\right]^{\ln n/\ln 2}}}

prod[n=1 până la ∞]{(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

Constanta Khinchin-Lévy

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

Constanta reciprocă Fibonacci

Ψ {\displaystyle \Psi }

∑ n = 1 ∞ 1 F n = 1 1 1 + 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

Rădăcina lui 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

Constanta Froda

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 la pătrat

π 2 {\displaystyle \pi ^{2}}

6 ∑ n = 1 ∞ 1 n 2 = 6 1 2 + 6 2 2 + 6 2 2 + 6 3 2 + 6 4 2 + ⋯ {\displaystyle 6\sum _{n=1}^{\infty }{\frac {1}{n^{2}}}={\frac {6}{6}{1^{2}}}+{\frac {6}{2^{2}}}+{\frac {6}{3^{2}}}+{\frac {6}{4^{2}}}+\cdots }

6 Sum[n=1 până la ∞]{1/n^2}

T

A002388

[9;1,6,1,2,47,1,8,1,1,2,2,1,1,8,3,1,10,5,...]

23.1406926327792690057290863679485474

Gelfond constant

e π {\displaystyle e^{\pi}}

∑ n = 0 ∞ π n n n ! = π 1 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 până la ∞]{(pi^n)/n!}

T

A039661

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