A221711 -id:A221711 - cp's OEIS frontend

A137245 Decimal expansion of Sum_{p prime} 1/(p * log p).

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

Offset: 1

R. J. Mathar, Mar 09 2008

Sum_{p prime} 1/(p^s * log p) equals this value here if s=1, equals A221711 if s=2, 0.22120334039... if s=3. See arXiv:0811.4739.
Erdős (1935) proved that for any sequence where no term divides another, the sum of 1/(x log x) is at most some constant C. He conjectures (1989) that C can be taken to be this constant 1.636..., that is, the primes maximize this sum. - Charles R Greathouse IV, Mar 26 2012 [The conjecture has been proved by Lichtman 2022. - Pontus von Brömssen, Jun 23 2022]
Note that sum 1/(p * log p) is almost (a tiny bit less than) 1 + 2/Pi = 1+A060294 = 1.63661977236758... (Why is it so close?) - Daniel Forgues, Mar 26 2012
Sum 1/(p * log p) is quite close to sum 1/n^2 = Pi^2/6 = 1.644934066... (Cf. David C. Ullrich, "Re: What is Sum(1/p log p)?" for why this is so; mentions A115563.) - Daniel Forgues, Aug 13 2012

			1.63661632335...

Henri Cohen, Number Theory, Volume II: Analytic and Modern Tools, GTM Vol. 240, Springer, 2007; see pp. 208-209.
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.27.2, p. 186.

Karim Belabas and Henri Cohen, Computation of sum_{p prime} 1/(p^s log(p)), PARI/GP script, 2020.
Henri Cohen, High-precision computation of Hardy-Littlewood constants, (1998).
Henri Cohen, High-precision computation of Hardy-Littlewood constants. [pdf copy, with permission]
P. Erdős, Note on sequences of integers no one of which is divisible by any other, J. London Math. Soc. 10 (1935), pp. 126-128, [DOI].
P. Erdős, Some problems and results on combinatorial number theory, Graph theory and its applications: East and West (Jinan, 1986), Ann. New York Acad. Sci., 576, pp. 132-145, New York Acad. Sci., New York, 1989.
Brady Haran and Jared Duker Lichtman, Primes and Primitive Sets, Numberphile video (2022).
Jared Duker Lichtman, Almost primes and the Banks-Martin conjecture, arXiv:1909.00804 [math.NT], 2019.
Jared Duker Lichtman, A proof of the Erdős primitive set conjecture, arXiv:2202.02384 [math.NT], 2022.
Jared Duker Lichtman, Proving the Erdős Primitive Set Conjecture, Oxford Mathematics YouTube video, 2022.
Jared Duker Lichtman, A proof of the Erdős primitive set conjecture, YouTube video from Combinatorial and additive number theory conference (CANT), 2022.
R. J. Mathar, Twenty digits of some integrals of the prime zeta function, arXiv:0811.4739 [math.NT], 2008-2009, table in Section 2.4.
David C. Ullrich, Re: What is Sum(1/p log p)?, posting in newsgroup sci.math.research, 11 Feb 2006.

Cf. A000040, A060294, A221711 (p squared), A115563, A319231 (log squared), A319232 (p and log squared), A354952.

digits = 105;
precision = digits + 10;
tmax = 500; (* integrand considered negligible beyond tmax *)
kmax = 500; (* f(k) considered negligible beyond kmax *)
InLogZeta[k_] := NIntegrate[Log[Zeta[t]], {t, k, tmax}, WorkingPrecision -> precision, MaxRecursion -> 20];
f[k_] := With[{mu = MoebiusMu[k]}, If[mu == 0, 0, (mu/k^2)*InLogZeta[k]]];
s = 0;
Do[s = s + f[k]; Print[k, " ", s], {k, 1, kmax}];
RealDigits[s][[1]][[1 ;; digits]] (* Jean-François Alcover, Feb 06 2021, updated Jun 22 2022 *)

\\ See Belabas, Cohen link. Run as SumEulerlog(1) after setting the required precision.

default(realprecision, 200); s=0; for(k=1, 500, s = s + moebius(k)/k^2 * intnum(x=k,[[1], 1], log(zeta(x))); print(s)); \\ Vaclav Kotesovec, Jun 12 2022

Equals Sum_{n>=1} 1/(A000040(n)*log A000040(n)).

More terms from Hugo Pfoertner, Feb 01 2020

More terms from Vaclav Kotesovec, Jun 12 2022

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

A319232 Decimal expansion of Sum_{p = prime} 1/(p*log p)^2.

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Sep 14 2018

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

			1/A016627^2 + 1/A016650^2 + 1/8.047189^2 + ... = 0.637056184074676....

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

Cf. A137245, A115563, A221711, A319231.

digits = 106; precision = digits + 10;
tmax = 500; (* integrand considered negligible beyond tmax *)
kmax = 300; (* f(k) considered negligible beyond kmax *)
InLogZeta[k_] := NIntegrate[(t - 2k) Log[Zeta[t]], {t, 2k, 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, 300, s = s + moebius(k)/k^3 * intnum(x=2*k,[[1], 1], (x-2*k)*log(zeta(x))); print(s)); \\ Vaclav Kotesovec, Jun 12 2022

More terms from Vaclav Kotesovec, Jun 12 2022

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

A354887 Decimal expansion of Sum_{primes p} log(log(p)) / (p*log(p)).

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, Jun 10 2022

			0.6410802156599846604833518891513999518913451...

Karim Belabas and Henri Cohen, Numerical Algorithms for Number Theory Using PariGP, American Mathematical Society, 2021, p.259-260.
Karim Belabas and Henri Cohen, Compute sum_{p prime} log(log(p))/(p^s log(p)), PARI/GP script, 2021.

Cf. A137245, A221711.

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.

A137250 Decimal expansion of the constant sum 1/(q*log(q)), summed over prime powers q > 1.

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Mar 09 2008

Evaluated from Sum_{m,k >= 1} A008683(k)*I(k*m)/k^2, where I(x) = Integral_{t=x..infinity} log zeta(t) dt is Cohen's underivative.

			2.0066664528310687...

Henri Cohen, Number Theory, Volume II: Analytic and Modern Tools, GTM Vol. 240, Springer, 2007; see pp. 208-209.

D. A. Clark, An upper bound of sum 1/(a_i log a_i) for quasi-primitive sequences, Comp. Math. Appl., 35 (1998), 105-109.
H. Cohen, High precision computation of Hardy-Littlewood constants, preprint, 1998.
Henri Cohen, High-precision computation of Hardy-Littlewood constants. [pdf copy, with permission]
R. J. Mathar, Twenty digits of some Integrals of the Prime Zeta Function, arXiv:0811.4739 [math.NT].

Cf. A137245, A221711.

default(realprecision, 200); su = 0; for(s=1, 400, su = su + sum(k=1, 500, moebius(k)/k^2 * intnum(x=s*k,[[1], 1], log(zeta(x))))/s; print(su)); \\ Vaclav Kotesovec, Jun 12 2022

Equals Sum_{n>=2} 1/(A000961(n)*log(A000961(n))).

Equals Sum_{p primes} -log(1-1/p)/log(p). - Vaclav Kotesovec, Jun 12 2022

8 more digits from R. J. Mathar, Dec 04 2008

More terms from Vaclav Kotesovec, Jun 12 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

A137245 Decimal expansion of Sum_{p prime} 1/(p * log p).

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

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A319232 Decimal expansion of Sum_{p = prime} 1/(p*log p)^2.

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

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A354887 Decimal expansion of Sum_{primes p} log(log(p)) / (p*log(p)).

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