A061444 -id:A061444 - cp's OEIS frontend

A075700 Decimal expansion of -zeta'(0).

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

Offset: 0

Benoit Cloitre, Oct 02 2002

The probability density function for the standard normal distribution is e^(-x^2/2 + zeta'(0)). - Rick L. Shepherd, Mar 08 2014

For every x > 0, PolyGamma(-2, x+1) - (PolyGamma(-2, x) + x*log(x) - x) equals this constant -zeta'(0), where polygamma functions of negative indices are defined for x > 0 as: PolyGamma(-1, x) = log(Gamma(x)), PolyGamma(-(n+1), x) = Integral_{t=0..x} PolyGamma(-n, x) dx, n >= 1. - Jianing Song, Apr 20 2021

			0.91893853320467274178032...

G. C. Greubel, Table of n, a(n) for n = 0..10000
J. Sondow and E. W. Weisstein, MathWorld: Wallis Formula.
Eric Weisstein's World of Mathematics, Log Gamma Function.
Eric Weisstein's World of Mathematics, Stirling's Approximation.
Wikipedia, Gamma function.
Wikipedia, Normal curve
Index entries for zeta function.

Cf. A019727, A061444, A257549.

SetDefaultRealField(RealField(100)); R:= RealField(); Log(2*Pi(R))/2; // G. C. Greubel, Oct 07 2018

evalf(log(2*Pi)/2,120); # Muniru A Asiru, Oct 08 2018

Log[Sqrt[2*Pi]] // RealDigits[#, 10, 104] & // First (* Jean-François Alcover, Apr 29 2013 *)

-zeta'(0) \\ Charles R Greathouse IV, Mar 28 2012

Equals log(2*Pi)/2 = A061444/2 = log(A019727).
Equals Integral_{x=0..1} log(Gamma(x)) dx. - Jean-François Alcover, Apr 29 2013
More generally, equals t-t*log(t)+Integral_{x=t..(t+1)} log(Gamma(x)) dx for any t>=0 (the Raabe formula). - Stanislav Sykora, May 14 2015
Equals lim_{k->oo} log(k!) + k - (k + 1/2)*log(k) (by Stirling's formula). - Amiram Eldar, Aug 21 2020

Normalized representation (leading zero and offset) R. J. Mathar, Jan 25 2009

A256127 Decimal expansion of the second Malmsten integral: Integer_{x >= 1} log(log(x))/(1 + x + x^2) dx, negated.

Original entry on oeis.org

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

Offset: 0

Iaroslav V. Blagouchine, Mar 15 2015

			-0.12632148170620903636522675325320239184424430946528...

G. C. Greubel, Table of n, a(n) for n = 0..5000
Iaroslav V. Blagouchine, Rediscovery of Malmsten's integrals, their evaluation by contour integration methods and some related results, The Ramanujan Journal, Volume 35, Issue 1, pp. 21-110, 2014, DOI: 10.1007/s11139-013-9528-5. PDF file
Wikipedia, Carl Malmsten

Cf. A115252 (first Malmsten integral), A256128 (third Malmsten integral) , A256129 (fourth Malmsten integral), A073005 (Gamma(1/3)), A256165 (log(Gamma(1/3))), A061444 (log(2*Pi)), A002391 (log 3), A002194 (sqrt 3).

evalf(Pi*(8*log(2*Pi) - 3*log(3) - 12*log(GAMMA(1/3)))/(6*sqrt(3)),120); # Vaclav Kotesovec, Mar 17 2015

RealDigits[Integrate[Log[Log[1/x]]/(1 + x + x^2), {x, 0, 1}], 10, 100][[1]] (* Alonso del Arte, Mar 16 2015 *)
RealDigits[Pi*(8*Log[2*Pi] - 3*Log[3] - 12*Log[Gamma[1/3]])/(6*Sqrt[3]),10,105][[1]] (* Vaclav Kotesovec, Mar 17 2015 *)

Pi*(8*log(2*Pi) - 3*log(3) - 12*log(gamma(1/3)))/(6*sqrt(3)) \\ Michel Marcus, Mar 18 2015

intnum(x=0, 1, log(log(1/x))/(1 + x + x^2))

intnum(x=1, oo, log(log(x))/(1 + x + x^2))

intnum(x=0, [oo, 1], log(x)/(1 + 2*cosh(x))) \\ Gheorghe Coserea, Sep 26 2018

Equals Integral_{x=0..1} log(log(1/x))/(1 + x + x^2) dx.
Equals Integral_{x>=0} log(x)/(1 + 2*cosh(x)) dx.
Equals Pi*(8*log(2*Pi) - 3*log(3) - 12*log(Gamma(1/3)))/(6*sqrt(3)).

A306243 Decimal expansion of Sum_{n>=2} log(n)/n!.

Original entry on oeis.org

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

Offset: 0

Rok Cestnik, Jan 31 2019

			0.6037828627914879884...

István Mező, Problem 11806, Problems and Solutions, The American Mathematical Monthly, Vol. 121, No. 10 (2014), p. 947; Parseval and Kummer, Solution to Problem 11806 by Omran Kouba, ibid., Vol. 123, No. 9 (2016), pp. 943-944.

Cf. A001620, A061444, A193424, A296301, A336730.

NSum[Log[n]/n!, {n, 2, Infinity}, WorkingPrecision -> 110,
  NSumTerms -> 100] // RealDigits[#, 10, 100] &

suminf(n=2, log(n)/n!) \\ Michel Marcus, Jan 31 2019

Equal to log(exp(1/2*log(2*exp(1/3*log(3*exp(1/4*log(4*exp(...)))))))).
Equals log(A296301). - Vaclav Kotesovec, Jun 22 2023
Equals Integral_{x=0..2*Pi} log(Gamma(x/(2*Pi))) * exp(cos(x)) * sin(x + sin(x)) dx - (e-1)*(log(2*Pi)+gamma), where gamma is Euler's constant (A001620) (Mező, 2014). - Amiram Eldar, Jan 25 2024
Equals Integral_{x=0..1} (exp(x) - 1)/(x*log(x)) - (exp(1) - 1)/log(x) dx. - Velin Yanev, Nov 29 2024

A345208 Decimal expansion of log(2*Pi) - gamma - 1, where gamma is Euler's constant (A001620).

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Jun 10 2021

The first two formulae (in the Formula section) are similar to the sum and integral lim_{n->oo} (1/n) * Sum_{k=1..n} frac(n/k) = Integral_{x=0..1} frac(1/x) dx = 1 - gamma (A153810).

The second raw moment of the distribution of the fractional part of 1/x, where x is chosen uniformly at random from (0, 1]. Since the expected value is 1 - gamma, the second central moment, or variance, is log(2*Pi) - gamma - 1 - (1 - gamma)^2 = log(2*Pi) - gamma^2 + gamma - 2 = 0.081914807503... and the standard deviation is sqrt(log(2*Pi) - gamma^2 + gamma - 2) = 0.2862076300...

			0.26066140150781262295414738272883284868063561133564...

Ovidiu Furdui, Limits, Series, and Fractional Part Integrals: Problems in Mathematical Analysis, New York: Springer, 2013. See Problem 3.42, pages 145 and 195.

Ovidiu Furdui, Solution to Problem U27, Mathematical Reflections, Vol. 6 (2006), pp. 27-28.
Ovidiu Furdui, Exotic fractional part integrals and Euler's constant, Analysis, Vol. 31, No. 3 (2011), pp. 249-257.
Mircea Ivan and Alexandru Lupaş, Problem 11206, The American Mathematical Monthly, Vol. 113, No. 2 (2006), p. 180; A Limit Involving Euler's Constant, Solution to problem 11206 by Richard A. Stong, ibid., Vol. 114, No. 10 (2007), pp. 928-929.
Albert F. S. Wong, Problem 1845, Mathematics Magazine, Vol. 83, No. 2 (2010), p. 150; Integrating a square-fractional-reciprocal function, Solution to problem 1845 by Allen Stenger, ibid., Vol. 84, No. 2 (2011), pp. 155-156.

Cf. A001008, A001620, A002805, A061444, A153810.

RealDigits[Log[2*Pi] - EulerGamma - 1, 10, 100][[1]]

Equals lim_{n->oo} (1/n) * Sum_{k=1..n} frac(n/k)^2, where frac(x) = x - floor(x) is the fractional part of x.
Equals Integral_{x=0..1} frac(1/x)^2 dx.
Equals 2 * Sum_{k>=2} (zeta(k)-1)/(k*(k+1)).
Equals A061444 - A001620 - 1.
Equals -2 * Sum_{k>=1} (H(k) - log(k) - gamma - 1/(2*k)), where H(k) = A001008(k)/A002805(k) is the k-th harmonic number (Furdui, 2013). - Amiram Eldar, Mar 26 2022

A216582 Decimal expansion of the logarithm of Pi to base 2.

Original entry on oeis.org

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

Offset: 1

Alonso del Arte, Sep 09 2012

			1.651496129472318798...

Jörg Arndt and Christoph Haenel, Pi Unleashed, New York: Springer (2001), p. 239.

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A002162, A020857, A053510, A053511, A061444, A094642.

SetDefaultRealField(RealField(100)); R:= RealField(); Log(Pi(R))/Log(2); // G. C. Greubel, Apr 08 2019

evalf(log[2](Pi)) ; # R. J. Mathar, Sep 11 2012

Mathematica
```
RealDigits[Log[2, Pi], 10, 105][[1]]
```

log(Pi)/log(2) \\ Michel Marcus, Mar 05 2019

log(pi,2).n(digits=100) # Jani Melik, Oct 05 2012

Log_2(Pi) = log(Pi) / log(2) = A053510 / A002162.

Equals (A061444 / A002162) - 1 = (A094642 / A002162) + 1. - John W. Nicholson, Mar 12 2019

A009763 a(n) is (n+1)!*(n+2)! times coefficient of x^n in (log(1-x))^-1.

Original entry on oeis.org

1, 1, 6, 76, 1620, 51780, 2310000, 136898496, 10393064640, 982930939200, 113269208976000, 15619762139984640, 2539231615282602240, 480507998223110457600, 104704722014993388288000, 26027184253285000629043200, 7320192187611052189440000000, 2312657526289162442074933248000

Offset: 0

Philippe Deléham, Apr 14 1997

nonn

Related to 'logarithmic numbers'.

Philippe Deléham, Letter to N. J. A. Sloane, Apr 14 1997
Wikipedia, Gregory Coefficients

Cf. A002206, A002207, A061444, A001620.

a := n -> local k; (-1)^n*(n+2)!*add(Stirling1(n+1, k)/(k+1), k = 0..n+1):
# Or:
ser := series(1/log(1-x), x, 20): seq((n+1)!*(n+2)!*coeff(ser, x, n), n = 0..17);
# Peter Luschny, Jun 23 2025

Table[(n+2)!*Abs[Sum[StirlingS1[n+1,k]/(k+1),{k,0,n+1}]],{n,0,20}] (* Vaclav Kotesovec, Aug 03 2014 *)
a[ n_] := If[n<0, 0, (n+2)!*(n+1)!*SeriesCoefficient[ 1/x + 1/Log[1-x], {x, 0, n}]]; (* Michael Somos, Jun 21 2025 *)

a(n)=local(A); if(n<0,0,n++; A=x/log(1-x+x^2*O(x^n)); n!*(n+1)!*polcoeff(A,n))

a(n) = (-1)^(n+1)*(n+1)!*(n+2)!*A002206/A002207(n).
log(2*Pi) = 1 + Sum_{n>0}{a(n)*(2n+1)/(((n+1)!)^2*n*(n+1)) = 1.83787706... = A061444. - Philippe Deléham, Jan 20 2004
Sum_{n>=0} a(n)/((n+1)*(n+1)!*(n+2)!) = Euler constant gamma = 0.5772156649... = A001620. - Philippe Deléham, Feb 26 2004
Sum_{n>0} a(n-1)/(n-1)! * x^n/n! = 1 + x/log(1-x). - Michael Somos, Jun 21 2025

Better description and more terms from Joe Keane (jgk(AT)jgk.org), Aug 13 2002

A122914 Decimal expansion of (1 + log(2*Pi))/2, the entropy of the standard normal distribution.

Original entry on oeis.org

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

Offset: 1

Joseph Biberstine (jrbibers(AT)indiana.edu), Sep 18 2006

For a normal distribution with standard deviation sigma, add log(sigma). - Stanislav Sykora, Jan 15 2017

			1.4189385332046727417803297364056176398613974736377834128171515404827656959...

Wikipedia, Normal distribution.

Partial quotients in A122915.

Cf. A061444, A075700.

Mathematica
```
RealDigits[(1 + Log[2 Pi])/2, 10, 80]
```

Equals (1 + log(2*Pi))/2 = 1/2 - A075700 = (1 + A061444)/2.
Equals -zeta(0) - zeta'(0). - Peter Luschny, May 16 2020
Equals 1 + G'(1), where G(x) is the Barnes G-function. - Amiram Eldar, Jun 08 2022

a(80) corrected by Georg Fischer, Jul 10 2021

A122915 Partial quotients of (1 + Log[2 Pi])/2, the entropy of the standard normal distribution.

Original entry on oeis.org

1, 2, 2, 1, 1, 2, 2, 8, 1, 8, 4, 1, 4, 1, 3, 18, 1, 7, 1, 4, 2, 4, 2, 1, 2, 1, 4, 1, 1, 17, 1, 1, 1, 23, 1, 1, 2, 28, 3, 2, 4, 1, 2, 3, 1, 39, 12, 2, 1, 1, 120, 1, 6, 1, 5, 1, 1, 1, 1, 1, 1, 4, 3, 1, 1, 5, 1, 5, 14, 1, 3, 3, 1, 5, 1, 12, 2, 7, 2, 1

Offset: 0

Joseph Biberstine (jrbibers(AT)indiana.edu), Sep 18 2006

a = 1/2 - A075700; a = (1 + A061444)/2.

Decimal expansion in A122914.

ContinuedFraction[(1 + Log[2 Pi])/2, 80]

a = (1 + Log[2 Pi])/2

A248859 Decimal expansion of log(sqrt(2*Pi))/e, a constant appearing in the asymptotic expansion of (n!)^(1/n).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Mar 03 2015

			0.3380585940662399023702794509615188742685137583402...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Steven R. Finch, Errata and Addenda to Mathematical Constants, arXiv:2001.00578 [math.HO], 2020-2021, p. 57.
Shafiqur Rahman and Leonard Giugiuc, Problem 4285, Crux Mathematicorum, Vol. 43, No. 9 (2017), pp. 399 and 401; Solution to Problem 4285, ibid., Vol. 44, No. 9 (2018), p. 395.

Cf. A001113, A019762, A061444, A075700 (log(sqrt(2*Pi))).

SetDefaultRealField(RealField(100)); R:= RealField(); Log(2*Pi(R))/(2*Exp(1)); // G. C. Greubel, Oct 07 2018

RealDigits[Log[Sqrt[2*Pi]]/E, 10, 105] // First

log(2*Pi)/2/exp(1) \\ Charles R Greathouse IV, Apr 20 2016

Equals lim_{n -> infinity} (n!)^(1/n) - n/e - log(n)/(2*e).

Equals A075700/A001113 = A061444/A019762. - Amiram Eldar, Apr 12 2022

A332538 Numerators of coefficients in a series for log(2 Pi).

Original entry on oeis.org

3, 1, 1, 7, 1, 43, 79, 717, 3481, 100189, 533077, 1777722593, 156155179, 74216302403, 15537618841, 11069240202341, 5762870563187, 2682308717818019, 927089189292457, 3726882116303677517, 35762248102620751, 1529769611935770520751, 1576432862602739502061

Offset: 0

N. J. A. Sloane, Feb 16 2020

Iaroslav V. Blagouchine and Marc-Antoine Coppo, A note on some constants related to the zeta-function and their relationship with the Gregory coefficients, arXiv:1703.08601 [math.NT], 2017. Also The Ramanujan Journal 47.2 (2018): 457-473. See Cor. 1 to Th. 2.

Cf. A002206, A002207, A332539.

Cf. A061444 (log(2*Pi)).

g[n_] := -(-1)^n*Sum[StirlingS1[n, j]/(j + 1), {j, 1, n}]/n!; Flatten[{3, Numerator[Table[Sum[g[k]*g[n + 1 - k], {k, 1, n}]/n, {n, 1, 30}]]}] (* Vaclav Kotesovec, Feb 16 2020 *)

The reference gives an explicit formula in terms of the Gregory numbers G_n = A002206/A002207.

More terms from Vaclav Kotesovec, Feb 16 2020

cp's OEIS Frontend

A075700 Decimal expansion of -zeta'(0).

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A256127 Decimal expansion of the second Malmsten integral: Integer_{x >= 1} log(log(x))/(1 + x + x^2) dx, negated.

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A306243 Decimal expansion of Sum_{n>=2} log(n)/n!.

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A345208 Decimal expansion of log(2*Pi) - gamma - 1, where gamma is Euler's constant (A001620).

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A216582 Decimal expansion of the logarithm of Pi to base 2.

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A009763 a(n) is (n+1)!*(n+2)! times coefficient of x^n in (log(1-x))^-1.

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