A131659 -id:A131659 - cp's OEIS frontend

A061444 Decimal expansion of log(2 * Pi).

1, 8, 3, 7, 8, 7, 7, 0, 6, 6, 4, 0, 9, 3, 4, 5, 4, 8, 3, 5, 6, 0, 6, 5, 9, 4, 7, 2, 8, 1, 1, 2, 3, 5, 2, 7, 9, 7, 2, 2, 7, 9, 4, 9, 4, 7, 2, 7, 5, 5, 6, 6, 8, 2, 5, 6, 3, 4, 3, 0, 3, 0, 8, 0, 9, 6, 5, 5, 3, 1, 3, 9, 1, 8, 5, 4, 5, 2, 0, 7, 9, 5, 3, 8, 9, 4, 8, 6, 5, 9, 7, 2, 7, 1, 9, 0, 8, 3, 9, 5, 2, 4

Offset: 1

Frank Ellermann, Jun 11 2001

Used in formulas for gamma(x), e.g., in Stirling's approximation for m!.
Also decimal expansion of zeta'(0)/zeta(0). - Benoit Cloitre, Sep 28 2002
The value of log(2*Pi) is close to 1 + Sum_{n>=2} log(zeta(n)) = 1.83067035427178011248.... - Arkadiusz Wesolowski, Jul 17 2011

			1.837877066409345483560659472811235279722794947275566825634303...

Harry J. Smith, Table of n, a(n) for n = 1..20000
Simon Plouffe, log(2*Pi) to 10000 digits
Simon Plouffe, Log(2*pi) to 2000 places

Cf. A002162, A053510, A094642, A131659.

RealDigits[N[Log[2*Pi], 100]][[1]] (* Arkadiusz Wesolowski, Aug 29 2011 *)

{ default(realprecision, 20080); x=log(2*Pi); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b061444.txt", n, " ", d)) } \\ Harry J. Smith, Jul 22 2009

Equals A002162 + A053510 = A131659 - A094642. - R. J. Mathar, Aug 27 2011

Equals 1 + Sum_{k>=1} zeta(2*k)/(k*(2*k + 1)). - Amiram Eldar, Aug 20 2020

A155968 Decimal expansion of (1/2)*log(Pi).

Original entry on oeis.org

5, 7, 2, 3, 6, 4, 9, 4, 2, 9, 2, 4, 7, 0, 0, 0, 8, 7, 0, 7, 1, 7, 1, 3, 6, 7, 5, 6, 7, 6, 5, 2, 9, 3, 5, 5, 8, 2, 3, 6, 4, 7, 4, 0, 6, 4, 5, 7, 6, 5, 5, 7, 8, 5, 7, 5, 6, 8, 1, 1, 5, 3, 5, 7, 3, 6, 0, 6, 8, 8, 8, 4, 9, 4, 2, 4, 1, 3, 0, 3, 9, 8, 9, 1, 8, 1, 1, 6, 3, 5, 1, 3, 7, 7, 4, 4, 8, 5, 3, 8, 5, 1, 0, 0, 4

Offset: 0

R. J. Mathar, Jan 31 2009

This sequence is also the decimal expansion of the logarithm of the Gamma-function at 1/2. - Iaroslav V. Blagouchine, Mar 20 2015

			0.572364942924700087071713675676529355823...

G. C. Greubel, Table of n, a(n) for n = 0..5000

Cf. A053510.

Maple
```
evalf(log(Pi)/2);
```

RealDigits[Log[Pi]/2,10,120][[1]] (* Harvey P. Dale, May 31 2015 *)

log(gamma(1/2)) \\ or \\ log(Pi)/2 \\ G. C. Greubel, Jan 16 2017

Equals A053510/2 = log(A002161) = A131659/4.

A335089 Decimal expansion of log(Pi^2/6).

Original entry on oeis.org

4, 9, 7, 7, 0, 0, 3, 0, 2, 4, 7, 0, 7, 4, 5, 3, 4, 7, 4, 7, 4, 3, 7, 7, 3, 4, 4, 3, 2, 5, 4, 1, 5, 1, 5, 0, 5, 7, 1, 5, 9, 8, 9, 3, 3, 6, 4, 7, 6, 1, 8, 4, 3, 7, 1, 7, 1, 8, 7, 1, 7, 9, 9, 8, 1, 3, 3, 8, 7, 6, 2, 4, 5, 8, 1, 3, 4, 8, 4, 7, 7, 0, 8, 7, 7, 6, 7, 4, 5, 8, 7, 4, 0, 8, 2, 8, 6, 3, 9, 0, 7, 4, 0, 4, 8, 1

Offset: 0

Terry D. Grant, Sep 11 2020

			Equals 1/(2^2) + 1/(3^2) + (1/(4^2))*(1/2) + 1/(5^2) + + 1/(7^2) + (1/(8^2))*(1/3) + ... = 0.4977003024707...

Mathematics Stack Exchange, Infinite series involving Von Mangoldt's function.
Grant Sanderson, What makes the natural log "natural"?, 3Blue1Brown video (2020).
Eric Weisstein's World of Mathematics, Mangold Function.
Eric Weisstein's World of Mathematics, Prime Zeta Function.

Cf. A000796, A013661, A053510, A131659, A016629.

Cf. A100995, A014963, A025474, A246655.

RealDigits[Log[Pi^2/6], 10, 120][[1]]
RealDigits[Sum[PrimeZetaP[2 k]/k, {k, 1, inf}], 10, 120][[1]]

log(Pi^2/6) \\ Michel Marcus, Sep 15 2020

Equals Sum_{k>=2} MangoldtLambda(k) / ((k^2)*log(k)).
Equals Sum_{k>=1} (1/k)*(1/(A246655(n)^2)) where k is the exponent of the prime power, A025474(n+1).
Equals Sum_{k>=1} primezeta(2*k)/k.
Equals 2*log(Pi) - log(6).
Equals log(zeta(2)) = log(A013661).

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A061444 Decimal expansion of log(2 * Pi).

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A155968 Decimal expansion of (1/2)*log(Pi).

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A335089 Decimal expansion of log(Pi^2/6).

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