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A085965 Decimal expansion of the prime zeta function at 5. 13
0, 3, 5, 7, 5, 5, 0, 1, 7, 4, 8, 3, 9, 2, 4, 2, 5, 7, 1, 3, 2, 8, 1, 8, 2, 4, 2, 5, 3, 8, 8, 5, 5, 7, 1, 1, 1, 3, 1, 6, 9, 7, 2, 7, 6, 7, 2, 6, 6, 5, 1, 3, 3, 1, 6, 9, 0, 0, 9, 2, 6, 7, 4, 8, 2, 3, 9, 7, 5, 8, 3, 4, 2, 7, 4, 7, 2, 7, 9, 3, 1, 3, 6, 6, 0, 7, 2, 8, 0, 6, 4, 7, 0, 3, 7, 6, 7, 9, 5, 0, 8, 9, 6, 3, 9 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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

REFERENCES

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

J. W. L. Glaisher, On the Sums of Inverse Powers of the Prime Numbers, Quart. J. Math. 25, 347-362, 1891.

LINKS

Jason Kimberley, Table of n, a(n) for n = 0..1702

Henri Cohen, High Precision Computation of Hardy-Littlewood Constants, Preprint, 1998.

Henri Cohen, High-precision computation of Hardy-Littlewood constants. [pdf copy, with permission]

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

R. J. Mathar, Series of reciprocal powers of k-almost primes, arXiv:0803.0900 [math.NT], 2008-2009. Table 1.

Eric Weisstein's World of Mathematics, Prime Zeta Function

FORMULA

P(5) = Sum_{p prime} 1/p^5 = Sum_{n>=1} mobius(n)*log(zeta(5*n))/n.

Equals 1/2^5 + A085994 + A086035. - R. J. Mathar, Jul 14 2012

Equals Sum_{k>=1} 1/A050997(k). - Amiram Eldar, Jul 27 2020

EXAMPLE

0.0357550174839242571328...

MAPLE

A085965:= proc(i) print(evalf(add(1/ithprime(k)^5, k=1..i), 100)); end:

A085965(100000); # Paolo P. Lava, May 29 2012

MATHEMATICA

s[n_] := s[n] = Sum[ MoebiusMu[k]*Log[Zeta[5*k]]/k, {k, 1, n}] // RealDigits[#, 10, 104]& // First // Prepend[#, 0]&; s[100]; s[n=200]; While[s[n] != s[n-100], n = n+100]; s[n] (* Jean-Fran├žois Alcover, Feb 14 2013, from 1st formula *)

RealDigits[ PrimeZetaP[ 5], 10, 111][[1]] (* Robert G. Wilson v, Sep 03 2014 *)

PROG

(MAGMA) R := RealField(106);

PrimeZeta := func<k, N | &+[R|MoebiusMu(n)/n*Log(ZetaFunction(R, k*n)): n in[1..N]]>;

[0] cat Reverse(IntegerToSequence(Floor(PrimeZeta(5, 69)*10^105)));

// Jason Kimberley, Dec 30 2016

(PARI) sumeulerrat(1/p, 5) \\ Hugo Pfoertner, Feb 03 2020

CROSSREFS

Decimal expansion of the prime zeta function: A085548 (at 2), A085541 (at 3), A085964 (at 4), this sequence (at 5), A085966 (at 6) to A085969 (at 9).

Cf. A013663, A050997, A242304.

Sequence in context: A324712 A279321 A254863 * A238205 A186702 A141710

Adjacent sequences:  A085962 A085963 A085964 * A085966 A085967 A085968

KEYWORD

cons,easy,nonn

AUTHOR

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

STATUS

approved

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Last modified March 1 03:32 EST 2021. Contains 341732 sequences. (Running on oeis4.)