A175478 -id:A175478 - cp's OEIS frontend

A253191 Decimal expansion of log(2)^2.

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

Offset: 0

Jean-François Alcover, Mar 24 2015

			0.480453013918201424667102526326664971730552951594545586866864...

David H. Bailey, Jonathan M. Borwein and Richard E. Crandall, On the Khintchine Constant, Mathematics of Computation, Vol. 66, No. 217 (1997), pp. 417-431, see p. 419; alternative link, p. 4.
Index entries for transcendental numbers

Cf. A001008, A002162, A002805, A175478.

Mathematica
```
RealDigits[Log[2]^2, 10, 103] // First
```

log(2)^2 \\ Charles R Greathouse IV, Apr 20 2016

Integral_{0..1} log(1-x^2)/(x*(1+x)) dx = -log(2)^2.
Integral_{0..1} log(log(1/x))/(x+sqrt(x)) dx = log(2)^2.
Equals Sum_{k>=1} H(k)/(2^k * (k+1)) = 2 * Sum_{k>=1} (-1)^(k+1) * H(k)/(k+1), where H(k) = A001008(k)/A002805(k) is the k-th harmonic number. - Amiram Eldar, Aug 05 2020
Equals Sum_{n >= 0} (-1)^n/(2^(n+1)*(n+1)^2*binomial(2*n+1,n)). See my entry in A002544 dated Apr 18 2017. Cf. A091476. - Peter Bala, Jan 30 2023
Equals 2*Integral_{x=-1..1} (abs(x)*log(x^2 + 1))/(x^2 + 1) dx. - Kritsada Moomuang, May 27 2025

A319231 Decimal expansion of Sum_{p = prime} 1/(p*log(p)^2).

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Sep 14 2018

Computed by expanding the formalism of arXiv:0811.4739 to double integrals over the Riemann zeta function.

			1/(2*A253191) + 1/(3*A175478) +1/(5*2.59029...) +1/(7*3.7865)+ ... = 1.52097043...

R. J. Mathar, Twenty digits of some integrals of the prime zeta function, arXiv:0811.4739 [math.NT], 2008-2018.

Cf. A137245, A115563, A221711, A319232, A354917.

digits = 105; precision = digits + 10;
tmax = 500; (* integrand considered negligible beyond tmax *)
kmax = 500; (* f(k) considered negligible beyond kmax *)
InLogZeta[k_] := NIntegrate[(t - k) Log[Zeta[t]], {t, k, tmax}, WorkingPrecision -> precision, MaxRecursion -> 20, AccuracyGoal -> precision];
f[k_] := With[{mu = MoebiusMu[k]}, If[mu == 0, 0, (mu/k^3)*InLogZeta[k]]];
s = 0;
Do[s = s + f[k]; Print[k, " ", s], {k, 1, kmax}];
RealDigits[s][[1]][[1 ;; digits]] (* Jean-François Alcover, Jun 21 2022, after Vaclav Kotesovec *)

default(realprecision, 200); s=0; for(k=1, 500, s=s+moebius(k)/k^3 * intnum(x=k,[[1], 1],(x-k)*log(zeta(x))); print(s)); \\ Vaclav Kotesovec, Jun 12 2022

More digits from Vaclav Kotesovec, Jun 12 2022

A175475 Decimal expansion of the Dickman function evaluated at 1/3.

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, May 25 2010

Density of the cube root-smooth numbers, see A090081. - Charles R Greathouse IV, Jul 14 2014

			F(1/3) = 0.04860838829113156690718...

David Broadhurst, Dickman polylogarithms and their constants arXiv:1004.0519 [math-ph], 2010.
K. Soundararajan, An asymptotic expansion related to the Dickman function, arXiv:1005.3494 [math.NT], 2010.

Cf. A002391, A072691, A175478.

N[1 - Log[3] + Log[3]^2/2 - Pi^2/12 + PolyLog[2, 1/3], 105] // RealDigits // First // Prepend[#, 0]& (* Jean-François Alcover, Feb 05 2013 *)

1-log(3)+log(3)^2/2-Pi^2/12+polylog(2,1/3) \\ Charles R Greathouse IV, Jul 14 2014

Equals 1 - log(3) + log^2(3)/2 - Pi^2/12 + Sum_{n>=1} 1/(n^2*3^n), where Sum_{n>=1} 1/(n^2*3^n) = 0.3662132299770634876167462976642627638...

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A253191 Decimal expansion of log(2)^2.

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A319231 Decimal expansion of Sum_{p = prime} 1/(p*log(p)^2).

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A175475 Decimal expansion of the Dickman function evaluated at 1/3.

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