A303493 -id:A303493 - cp's OEIS frontend

A136271 Decimal expansion of sum log(p)/p^2 over the primes p=2,3,5,7,11,...

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

Offset: 0

R. J. Mathar, Mar 09 2008

The negative first derivative of the prime zeta function at 2.

			0.493091109368764462197826205056491258055588126346468290713327120321336...

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

Henri Cohen, High-precision computation of Hardy-Littlewood constants, preprint 1998.
Henri Cohen, High-precision computation of Hardy-Littlewood constants. [pdf copy, with permission]
Tengiz O. Gogoberidze, Baker's dozen digits of two sums involving reciprocal products of an integer and its greatest prime factor, arXiv:2407.12047 [math.GM], 2024. See p. 2.
R. J. Mathar, Series of reciprocal powers of k-almost primes, arXiv:0803.0900 [math.NT], Table 3.
J. B. Rosser, L. Schoenfeld, Approximate formulas for some functions of prime numbers, Ill. J. Math. 6 (1) (1962) 64-94, Table IV

Cf. A303493-A303499, A360094, A360095.

RealDigits[PrimeZetaP'[2], 10, 104] // First (* Jean-François Alcover, Sep 11 2015 *)

Equals Sum_{n >= 1} log(A000040(n))/A001248(n).

Removed 6 digits where preprints disagree. Added zero to an A-number in formula. Corrected Cohen preprint year. Added 2nd link. - R. J. Mathar, Nov 27 2008

More digits from Jean-François Alcover, Sep 11 2015

A303499 Decimal expansion of Sum_{p prime} log(p)/p^9.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Apr 25 2018

The negated first derivative of the Prime Zeta function at 9.

			0.0014104919214245312915541964563081999779016571316934961928365008287798...

Eric Weisstein's MathWorld, Prime Zeta Function
Wikipedia, Prime Zeta Function

Decimal expansion of Sum_{p prime} log(p)/p^k: A136271 (k=2), A303493 (k=3), A303494 (k=4), A303495 (k=5), A303496 (k=6), A303497 (k=7), A303498 (k=8), A303499 (k=9).

RealDigits[PrimeZetaP'[9], 10, 103][[1]]

suminf(n=1, p=prime(n); log(p)/p^9) \\ Michel Marcus, Apr 25 2018

A303494 Decimal expansion of Sum_{p prime} log(p)/p^4.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Apr 25 2018

The negated first derivative of the Prime Zeta function at 4.

			0.060607633350770063392230983709713378406382877461259843991127681734...

R. J. Mathar, Series of reciprocal powers of k-almost primes, arXiv:0803.0900 (2008) Table 3, s=4.
J. B. Rosser, L. Schoenfeld, Approximate formulas for some functions of prime numbers, Ill. J. Math. 6 (1) (1962) 64-94, Table IV
Eric Weisstein's MathWorld, Prime Zeta Function
Wikipedia, Prime Zeta Function

Decimal expansion of Sum_{p prime} log(p)/p^k: A136271 (k=2), A303493 (k=3), A303494 (k=4), A303495 (k=5), A303496 (k=6), A303497 (k=7), A303498 (k=8), A303499 (k=9).

RealDigits[PrimeZetaP'[4], 10, 100][[1]]

A303495 Decimal expansion of Sum_{p prime} log(p)/p^5.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Apr 25 2018

The negated first derivative of the prime zeta function at 5.

			0.0268386012767983574221875132924501599433301495535582281248198036...

R. J. Mathar, Series of reciprocal powers of k-almost primes arXiv:0803.0900, Table 3.
Eric Weisstein's MathWorld, Prime Zeta Function
Wikipedia, Prime Zeta Function

Decimal expansion of Sum_{p prime} log(p)/p^k: A136271 (k=2), A303493 (k=3), A303494 (k=4), A303495 (k=5), A303496 (k=6), A303497 (k=7), A303498 (k=8), A303499 (k=9).

Mathematica
```
RealDigits[PrimeZetaP'[5], 10, 99][[1]]
```

A303496 Decimal expansion of Sum_{p prime} log(p)/p^6.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Apr 25 2018

The negated first derivative of the prime zeta function at 6.

			0.012459080722799991527027792774689970040911350471575875874109334035...

R. J. Mathar, Series of reciprocal powers of k-almost primes arXiv:0803.0900, Table 3 s=6.
Eric Weisstein's MathWorld, Prime Zeta Function
Wikipedia, Prime Zeta Function

Decimal expansion of Sum_{p prime} log(p)/p^k: A136271 (k=2), A303493 (k=3), A303494 (k=4), A303495 (k=5), A303496 (k=6), A303497 (k=7), A303498 (k=8), A303499 (k=9).

RealDigits[PrimeZetaP'[6], 10, 100][[1]]

A303497 Decimal expansion of Sum_{p prime} log(p)/p^7.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Apr 25 2018

The negated first derivative of the Prime Zeta function at 7.

			0.00594068903914819614255059282901660901936818950592935107516681316988898...

Eric Weisstein's MathWorld, Prime Zeta Function
Wikipedia, Prime Zeta Function

Decimal expansion of Sum_{p prime} log(p)/p^k: A136271 (k=2), A303493 (k=3), A303494 (k=4), A303495 (k=5), A303496 (k=6), A303497 (k=7), A303498 (k=8), A303499 (k=9).

RealDigits[PrimeZetaP'[7], 10, 103][[1]]

A303498 Decimal expansion of Sum_{p prime} log(p)/p^8.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Apr 25 2018

The negated first derivative of the Prime Zeta function at 8.

			0.00287952470872924739134602842385733406499898376167586584106761828618532...

Eric Weisstein's MathWorld, Prime Zeta Function
Wikipedia, Prime Zeta Function

Decimal expansion of Sum_{p prime} log(p)/p^k: A136271 (k=2), A303493 (k=3), A303494 (k=4), A303495 (k=5), A303496 (k=6), A303497 (k=7), A303498 (k=8), A303499 (k=9).

RealDigits[PrimeZetaP'[8], 10, 103][[1]]

suminf(n=1, p=prime(n); log(p)/p^8) \\ Michel Marcus, Apr 25 2018

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.

cp's OEIS Frontend

A136271 Decimal expansion of sum log(p)/p^2 over the primes p=2,3,5,7,11,...

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A303499 Decimal expansion of Sum_{p prime} log(p)/p^9.

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A303494 Decimal expansion of Sum_{p prime} log(p)/p^4.

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A303495 Decimal expansion of Sum_{p prime} log(p)/p^5.

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A303496 Decimal expansion of Sum_{p prime} log(p)/p^6.

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A303497 Decimal expansion of Sum_{p prime} log(p)/p^7.

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A303498 Decimal expansion of Sum_{p prime} log(p)/p^8.

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

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