A354917 -id:A354917 - cp's OEIS frontend

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

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

A145419 Decimal expansion of Sum_{k>=2} 1/(k*(log k)^3).

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Feb 08 2009

Cubic 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.

Theorem: Bertrand series Sum_{n>=2} 1/(n*log(n)^q) is convergent iff q > 1 (for q = 2, 4, 5 see respectively A115563, A145420, A145421). - Bernard Schott, Oct 23 2021

			2.0658865388841352509031422416437738180869752069383...

R. J. Mathar, The series limit of sum_k 1/[k log k (log log k)^2], arXiv:0902.0789 [math.NA], 2009-2021, constant D^(3).
Wikipédia, Série de Bertrand (in French).

Cf. A115563, A145420, A145421, A354917.

digits = 50; NSum[ 1/(n*Log[n]^3), {n, 2, Infinity}, NSumTerms -> 10000, WorkingPrecision -> digits + 10] // RealDigits[#, 10, digits] & // First (* Jean-François Alcover, Feb 11 2013 *)
alfa = 3; maxiter = 20; nn = 10000; 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 11 2022 *)

More terms from Jean-François Alcover, Feb 11 2013

More digits from Vaclav Kotesovec, Jun 11 2022

A363368 Decimal expansion of Sum_{primes p} 1/(plog(p)log(log(p))).

Original entry on oeis.org

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

Offset: 1

Artur Jasinski, Jun 11 2023

Value computed and communicated by Bill Allombert and confirmed by Pascal Sebah.

			1.9069738480349544...

Cf. A077761, A086242, A137245, A138312, A221711, A303493, A319231, A319232, A354887, A354917, A354952, A361972, A362533.

/* author Bill Allombert */
\p150
pz(x, ex=0)=
{
my(s=bitprecision(x));
my(B=s/real(polcoef(x, 0))+ex);
sum(n=1, B, my(a=moebius(n)); if(a!=0, a*log(zeta(n*x))/n));
}
my(P=primes([2, 61])); intnum(x=1, [oo, log(67)], (pz(x)-vecsum([p^-x|p<-P]))*intnum(s=0, [oo, 1], (x-1)^s/gamma(1+s))) + vecsum([1/p/log(p)/log(log(p))|p<-P])

A361089 a(n) = smallest integer x such that Sum_{k = 2..x} 1/(k*log(log(k))) > n.

Original entry on oeis.org

3, 5, 8, 21, 76, 389, 2679, 23969, 269777, 3717613, 61326301, 1188642478, 26651213526, 682263659097, 19720607003199, 637490095320530, 22857266906194526, 902495758030572213, 38993221443197045348, 1833273720522384358862

Offset: 2

Artur Jasinski, Jun 11 2023

nonn

Because lim_{x->oo} (Sum_{k=2..x} 1 / (k*log(log(k)))) - li(log(x)) = 2.7977647035208... (see A363078) then a(n) = round(w) where w is the solution of the equation li(log(w)) + 2.7977647035208... = n.

			a(2) = 3 because Sum_{k=2..3} 1/(k*log(log(k))) = 2.18008755... > 2 and Sum_{k=2..2} 1/(k*log(log(k))) = -1.364208386450... < 2.
a(7) = 389 because Sum_{k=2..389} 1/(k*log(log(k))) = 7.000345... > 7 and Sum_{k=2..388} 1/(k*log(log(k))) = 6.99890560988... < 7.

Pascal Sebah, Table of n, a(n) for n = 2..35

Cf. A077761, A086242, A137245, A138312, A221711, A303493, A319231, A319232, A354887, A354917, A354952, A361972, A362533, A363368, A363078.

(*slow procedure*)
lim = 2; sum = 0; aa = {}; Do[sum = sum + N[1/(k Log[Log[k]]), 100];
 If[sum >= lim, AppendTo[aa, k]; Print[{lim, sum, k}];
  lim = lim + 1], {k, 2, 269777}];aa
(*quick procedure *)
aa = {3}; cons = 2.79776470352080492766050456553352884330850083202326989577856315;
Do[ww = w /. NSolve[LogIntegral[Log[w]] + cons == n, w];
 AppendTo[aa, Round[ww][[1]]], {n, 3, 21}]; aa

For n >= 3, a(n) = round(w) where w is the solution of the equation li(log(w)) + 2.7977647035208... = n.

A363078 Decimal expansion of lim_{x->oo} (Sum_{k=2..x} 1 / (k*log(log(k)))) - li(log(x)).

Original entry on oeis.org

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

Offset: 1

Artur Jasinski, Jun 11 2023

Value computed and communicated by Pascal Sebah.

For the smallest integer x such that Sum_{k = 2..x} 1/(k*log(log(k))) > n see A361089.

			2.7977647035208...

Cf. A077761, A086242, A137245, A138312, A221711, A303493, A319231, A319232, A354887, A354917, A354952, A361972, A362533, A363368, A361089.

A354953 Decimal expansion of Sum_{p = primes} 1 / (p * log(p)^5).

Original entry on oeis.org

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

Offset: 1

Vaclav Kotesovec, Jun 13 2022

			3.359898760127253088364274368063313570407472689603469004194863140645872...

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

Cf. A137245, A319231, A354917, A354954.

digits = 105; precision = digits + 15;
tmax = 400; (* integrand considered negligible beyond tmax *)
kmax = 400; (* f(k) considered negligible beyond kmax *)
InLogZeta[k_] := NIntegrate[(t-k)^4 Log[Zeta[t]], {t, k, tmax},
  WorkingPrecision -> precision, MaxRecursion -> 20,
  AccuracyGoal -> precision];
f[k_] := With[{mu = MoebiusMu[k]}, If[mu==0, 0, (mu/(4! k^6))* InLogZeta[k]]];
s = 0; Do[s = s + f[k]; Print[k, " ", s], {k, 1, kmax}];
RealDigits[s][[1]][[1 ;; digits]] (* Jean-François Alcover, Jun 23 2022 *)

default(realprecision, 200); s=0; for(k=1, 500, s = s + moebius(k)/(4!*k^6) * intnum(x=k,[[1], 1], (x-k)^4 * log(zeta(x))); print(s));

Last 5 digits corrected by Vaclav Kotesovec, Jun 22 2022, following a suggestion from Jean-François Alcover

A354954 Decimal expansion of Sum_{p = primes} 1 / (p * log(p)^4).

Original entry on oeis.org

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

Offset: 1

Vaclav Kotesovec, Jun 13 2022

			2.443227043544410188729683297369734576461453087740400428664651485267350...

R. J. Mathar, Twenty digits of some integrals of the prime zeta function, arXiv:0811.4739 (2008-2009).

Cf. A137245, A319231, A354917, A354953.

default(realprecision, 200); s=0; for(k=1, 500, s = s + moebius(k)/(6*k^5) * intnum(x=k,[[1], 1], (x-k)^3 * log(zeta(x))); print(s));

Last 2 digits corrected by Vaclav Kotesovec, Jun 22 2022

A366249 Decimal expansion of lim_{x->oo} (Sum_{primes p<=x} 1/(p*log(log(p)))) - log(log(log(x))).

Original entry on oeis.org

2, 9, 3, 8, 3, 2, 9, 0, 1

Offset: 1

Artur Jasinski, Oct 05 2023

Value computed and communicated by Pascal Sebah.

			2.93832901...

Cf. A077761, A086242, A137245, A138312, A221711, A303493, A319231, A319232, A354887, A354917, A354952, A361972, A362533, A363078, A363368.

cp's OEIS Frontend

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

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A145419 Decimal expansion of Sum_{k>=2} 1/(k*(log k)^3).

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A363368 Decimal expansion of Sum_{primes p} 1/(p*log(p)*log(log(p))).

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A361089 a(n) = smallest integer x such that Sum_{k = 2..x} 1/(k*log(log(k))) > n.

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A363078 Decimal expansion of lim_{x->oo} (Sum_{k=2..x} 1 / (k*log(log(k)))) - li(log(x)).

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A354953 Decimal expansion of Sum_{p = primes} 1 / (p * log(p)^5).

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A354954 Decimal expansion of Sum_{p = primes} 1 / (p * log(p)^4).

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A366249 Decimal expansion of lim_{x->oo} (Sum_{primes p<=x} 1/(p*log(log(p)))) - log(log(log(x))).

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A363368 Decimal expansion of Sum_{primes p} 1/(plog(p)log(log(p))).