A145419 -id:A145419 - cp's OEIS frontend

A115563 Decimal expansion of Sum_{n>=2} 1/(n*log(n)^2).

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

Offset: 1

Pierre CAMI, Mar 11 2006

Sum_{n>1} 1/(n*log(n)^2) is a tiny bit greater than (zeta(2))^(3/2) = (Pi^2 / 6)^(3/2) = 2.109709908063657.... - Daniel Forgues, Mar 30 2012
From Bernard Schott, Oct 03 2021: (Start)
Theorem: Bertrand series Sum_{n>=2} 1/(n*log(n)^q) is convergent iff q > 1 (for q = 3, 4, 5 see respectively A145419, A145420, A145421).
As H(n) ~ log(n), compare with A347145. (End)

			2.10974280123689197447925719761655132638553198439474202264991560319281...

John V. Baxley, Euler's constant, Taylor's formula, and slowly converging series, Math. Mag. 65 (1992), 302-313.
Bart Braden, Calculating sums of infinite series, Am. Math. Monthly 99 (1992) 649-655.
David Broadhurst, Re: need help about 2 constants, primeforum, Mar 10 2006.
Pierre CAMI, David Broadhurst, Need help about 2 constants, digest of 3 messages in primeform Yahoo group, Mar 10, 2006. [Cached copy]
Rick Kreminski, Using Simpson's rule to approximate sums of infinite series, College Math. J. 28 (1997), 368-376.
R. J. Mathar, The series limit of sum_k 1/[k*log k *(log log k)^2], arXiv:0902.0789 [math.NA], 2009, App. A.
S.O.S. Math, Bertrand Series.
Eric Weisstein's World of Mathematics, Convergent Series
Wikipédia, Série de Bertrand (in French).

Cf. A145419, A145420, A145421, A363632, A363633.
Cf. A137245, A257812. A097906 is a similar sum.
Cf. A118582, A347145.

digits = 150; NSum[1/(n*Log[n]^2), {n, 2, Infinity}, NSumTerms -> 200000, WorkingPrecision -> digits + 5, Method -> {"EulerMaclaurin", Method -> {"NIntegrate", "MaxRecursion" -> 20}}] (* Vaclav Kotesovec, Mar 01 2016, after Jean-François Alcover *)
maxiter = 20; nn = 10000; alfa = 2; bas = Sum[1/(k*Log[k]^alfa), {k, 2, nn}] + 1/((alfa - 1)*Log[nn + 1/2]^(alfa - 1)); sub = 0; Do[sub = sub + 1/4^s/(2*s + 1)! * NSum[(D[1/(x*Log[x]^alfa), {x, 2 s}]) /. x -> k, {k, nn + 1, Infinity}, WorkingPrecision -> 120, NSumTerms -> 100000, PrecisionGoal -> 120, Method -> {"NIntegrate", "MaxRecursion" -> 100}]; Print[bas - sub], {s, 1, maxiter}] (* Vaclav Kotesovec, Jun 12 2022 *)

Removed incorrect speculations about relations to A097906 - R. J. Mathar, Oct 14 2010
More terms from Robert G. Wilson v, Dec 12 2012
Corrected a(55) and beyond, Vaclav Kotesovec, Mar 01 2016

A354917 Decimal expansion of Sum_{p = primes} 1 / (p * log(p)^3).

Original entry on oeis.org

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

Offset: 1

Vaclav Kotesovec, Jun 12 2022

			1.8461474193664495...

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

Cf. A137245, A145419, A221711, A319231, A319232.

digits = 105; precision = digits + 15;
tmax = 500; (* integrand considered negligible beyond tmax *)
kmax = 500; (* f(k) considered negligible beyond kmax *)
InLogZeta[k_] := NIntegrate[(t - k)^2 Log[Zeta[t]], {t, k, tmax}, WorkingPrecision -> precision, MaxRecursion -> 20, AccuracyGoal -> precision];
f[k_] := With[{mu = MoebiusMu[k]}, If[mu == 0, 0, (mu/(2 k^4))*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)/(2*k^4) * intnum(x=k,[[1], 1], (x-k)^2 * log(zeta(x))); print(s));

Last digit corrected by Jean-François Alcover and confirmed by Vaclav Kotesovec, Jun 22 2022

A145420 Decimal expansion of Sum_{k>=2} 1/(k*(log k)^4).

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Feb 08 2009

Quartic analog of A115563. Evaluated by direct summation of the first 160 terms and accumulating the remainder with the 5 nontrivial terms in the Euler-Maclaurin expansion.

Bertrand series Sum_{n>=2} 1/(n*log(n)^q) is convergent iff q > 1. - Bernard Schott, Jan 22 2022

			2.5591197429867314185720209703107629336191781563668...

R. J. Mathar, The series limit of sum_k 1/[k log k (log log k)^2], arXiv:0902.0789 [math.NA], 2009-2021, last sentence.
Wikipédia, Série de Bertrand (in French).

Cf. A115563 (q=2), A145419 (q=3), A145421 (q=5).

(* Computation needs a few minutes *) digits = 105; NSum[ 1/(n*Log[n]^4), {n, 2, Infinity}, NSumTerms -> 800000, WorkingPrecision -> digits + 5, Method -> {"EulerMaclaurin", Method -> {"NIntegrate", "MaxRecursion" -> 10}}] // RealDigits[#, 10, digits] & // First (* Jean-François Alcover, Feb 12 2013 *)

More terms from Jean-François Alcover, Feb 12 2013

A145421 Decimal expansion of Sum_{k>=2} 1/(k*(log k)^5).

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Feb 08 2009

Quintic analog of A115563. Evaluated by direct summation of the first 160 terms and accumulating the remainder with the 5 nontrivial terms in the Euler-Maclaurin expansion.

Bertrand series Sum_{n>=2} 1/(n*log(n)^q) is convergent iff q > 1. - Bernard Schott, Feb 08 2022

			3.4298162600230560650224115856558605404523762001207...

Wikipédia, Série de Bertrand (in French).

Cf. A115563 (q=2), A145419 (q=3), A145420 (q=4).

(* Computation needs a few minutes *) digits = 105; NSum[ 1/(n*Log[n]^5), {n, 2, Infinity}, NSumTerms -> 1500000, WorkingPrecision -> digits + 10, Method -> {"EulerMaclaurin", Method -> {"NIntegrate", "MaxRecursion" -> 10}}] // RealDigits[#, 10, digits] & // First (* Jean-François Alcover, Feb 12 2013 *)

More terms from Jean-François Alcover, Feb 12 2013

A363632 Decimal expansion of Sum_{k>=2} 1/(k* log(k)^(3/2)).

Original entry on oeis.org

2, 9, 3, 7, 6, 6, 3, 6, 3, 7, 9, 0, 1, 2, 3, 1, 7, 7

Offset: 1

R. J. Mathar, Jun 12 2023

			2.93766363790123177...

Cf. A115563 (expo 2), A363633 (expo 5/2), A145419 (expo 3), A145420 (expo 4).

A363633 Decimal expansion of Sum_{k>=2} 1/(k* log(k)^(5/2)).

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Jun 12 2023

			1.98346354715623939118...

Cf. A363632 (expo 3/2), A115563 (expo 2), A145419 (expo 3), A145420 (expo 4).

cp's OEIS Frontend

A115563 Decimal expansion of Sum_{n>=2} 1/(n*log(n)^2).

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A354917 Decimal expansion of Sum_{p = primes} 1 / (p * log(p)^3).

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A145420 Decimal expansion of Sum_{k>=2} 1/(k*(log k)^4).

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A145421 Decimal expansion of Sum_{k>=2} 1/(k*(log k)^5).

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A363632 Decimal expansion of Sum_{k>=2} 1/(k* log(k)^(3/2)).

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A363633 Decimal expansion of Sum_{k>=2} 1/(k* log(k)^(5/2)).

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