A136271 -id:A136271 - cp's OEIS frontend

A085548 Decimal expansion of the prime zeta function at 2: Sum_{p prime} 1/p^2.

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

Offset: 0

Cino Hilliard, Jul 03 2003

Mathar's Table 1 (cited below) lists expansions of the prime zeta function at integers s in 10..39. - Jason Kimberley, Jan 05 2017

			0.4522474200410654985065... = 1/2^2 + 1/3^2 + 1/5^2 +1/7^2 + 1/11^2 + 1/13^2 + ...

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, pp. 94-98.

Jason Kimberley, Table of n, a(n) for n = 0..1093
Henri Cohen, High Precision Computation of Hardy-Littlewood Constants, Preprint, 1998.
Henri Cohen, High-precision computation of Hardy-Littlewood constants. [pdf copy, with permission]
Persi Diaconis, Frederick Mosteller, and Hironari Onishi, Second-order terms for the variances and covariances of the number of prime factors-including the square free case, J. Number Theory 9 (1977), no. 2, 187--202. MR0434991 (55 #7953).
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 171 and 190.
J. W. L. Glaisher, On the Sums of Inverse Powers of the Prime Numbers, Quart. J. Math. 25, 347-362, 1891.
Xavier Gourdon and Pascal Sebah, Some Constants from Number theory.
Lajos Hajdu and Rob Tijdeman, Integers represented by Lucas sequences, arXiv:2408.04982 [math.NT], 2024. See p. 16.
Shanta Laishram and Florian Luca, Rectangles Of Nonvisible Lattice Points, J. Int. Seq. 18 (2015), Article 15.10.8, Theorem 1.
Jon Lee, Joseph Paat, Ingo Stallknecht, and Luze Xu, Polynomial upper bounds on the number of differing columns of Delta-modular integer programs, arXiv:2105.08160 [math.OC], 2021, see page 23.
R. J. Mathar, Series of reciprocal powers of k-almost primes, arXiv:0803.0900 [math.NT], 2008-2009. Table 1.
Gerhard Niklasch and Pieter Moree, Some number-theoretical constants. [Cached copy]
Michael Ian Shamos, Property Enumerators and a Partial Sum Theorem, 2011; alternative link.
Hanson Smith, Ramification in the Division Fields of Elliptic Curves and an Application to Sporadic Points on Modular Curves, arXiv:1810.04809 [math.NT], 2018.
Hanson Smith, Ramification in Division Fields and Sporadic Points on Modular Curves, U. Conn. (2020).
Eric Weisstein's World of Mathematics, Distinct Prime Factors.
Eric Weisstein's World of Mathematics, Prime Sums.
Eric Weisstein's World of Mathematics, Prime Zeta Function.
Wikipedia, Prime Zeta Function.
Index to constants which are prime zeta sums {2,0,0}

Decimal expansion of the prime zeta function: this sequence (at 2), A085541 (at 3), A085964 (at 4) to A085969 (at 9).
Cf. A136271 (derivative), A117543 (semiprimes), A222056, A209329, A124012.
Cf. A000720, A001248, A013661, A078437, A242301.

R := RealField(106);
PrimeZeta := func;
Reverse(IntegerToSequence(Floor(PrimeZeta(2,173)*10^105)));
// Jason Kimberley, Dec 30 2016

RealDigits[PrimeZetaP[2], 10, 105][[1]]  (* Jean-François Alcover, Jun 24 2011, updated May 06 2021 *)

recip2(n) = { v=0; p=1; forprime(y=2,n, v=v+1./y^2; ); print(v) }

eps()=my(p=default(realprecision)); precision(2.>>(32*ceil(p*38539962/371253907)),9)
lm=lambertw(log(4)/eps())\log(4);
sum(k=1,lm, moebius(k)/k*log(abs(zeta(2*k)))) \\ Charles R Greathouse IV, Jul 19 2013

sumeulerrat(1/p,2) \\ Hugo Pfoertner, Feb 03 2020

P(2) = Sum_{p prime} 1/p^2 = Sum_{n>=1} mobius(n)*log(zeta(2*n))/n. - Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Jul 06 2003
Equals A085991 + A086032 + 1/4. - R. J. Mathar, Jul 22 2010
Equals Sum_{k>=1} 1/A001248(k). - Amiram Eldar, Jul 27 2020
Equals Sum_{k>=2} pi(k)*(2*k+1)/(k^2*(k+1)^2), where pi(k) = A000720(k) (Shamos, 2011, p. 9). - Amiram Eldar, Mar 12 2024

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Jul 06 2003

Offset corrected by R. J. Mathar, Feb 05 2009

A303493 Decimal expansion of Sum_{p prime} log(p)/p^3.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Apr 25 2018

The negated first derivative of the Prime Zeta function at 3.

			0.150757555543950422179836516365342919575501161530689331818797676...

R. Mathar, Series of reciprocal powers of k-almost primes, arXiv:0803.0900 [math.NT], 2008-2009; Table 3, s=3.
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'[3], 10, 100][[1]]

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

A358789 Decimal expansion of Sum_{p prime, p>=3} (-1)^((p-1)/2)*log(p)/p, negated.

Original entry on oeis.org

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

Offset: 0

Artur Jasinski, Jan 03 2023

Sum_{p prime} log(p)/p is divergent.

			-0.54568127279512790148953238338...

Cf. A086239, A091812, A136271, A354295.

alfa[s_]:= 1/(1 + 1/2^s) * DirichletBeta[s] * Zeta[s] / Zeta[2*s]; beta[s_]:= (1 - 1/2^s) * Zeta[s] / DirichletBeta[s]; Do[Print[N[-1/2*Sum[MoebiusMu[2*n + 1]/(2*n + 1) * Limit[D[Log[alfa[(2*n + 1)*s]/beta[(2*n + 1)*s]], s], s -> 1], {n, 0, m}], 120]], {m, 20, 200, 20}] (* Vaclav Kotesovec, Jan 25 2023 *)

Limit_{N->oo} ((Sum_{p<=N prime == 3 (mod 4)} log(p)/p) - (Sum_{p<=N prime == 1 (mod 4)} log(p)/p)).

A360094 Decimal expansion of Sum_{p primes, p == 1 mod 4} log(p)/p^2.

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, Jan 25 2023

			0.107359545297113077138450382009121901166339396912637779372659580780278...

X. Gourdon and P. Sebah, Some Constants from Number theory.

Cf. A085548, A086032, A086239, A136271, A358789, A360095.

alfa[s_]:= 1/(1 + 1/2^s) * DirichletBeta[s] * Zeta[s] / Zeta[2*s]; Do[Print[N[-1/2*Sum[MoebiusMu[2*n + 1]/(2*n + 1) * D[Log[alfa[(2*n + 1)*s]], s] /. s->2, {n, 0, m}], 120]], {m, 10, 100, 10}]

Equals A136271 - A360095 - log(2)/4.

cp's OEIS Frontend

A085548 Decimal expansion of the prime zeta function at 2: Sum_{p prime} 1/p^2.

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A303493 Decimal expansion of Sum_{p prime} log(p)/p^3.

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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.