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A085965
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Decimal expansion of the prime zeta function at 5.
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13
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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
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OFFSET
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0,2
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COMMENTS
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Mathar's Table 1 (cited below) lists expansions of the prime zeta function at integers s in 10..39. - Jason Kimberley, Jan 05 2017
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REFERENCES
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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.
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LINKS
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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
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FORMULA
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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
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EXAMPLE
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0.0357550174839242571328...
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MAPLE
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A085965:= proc(i) print(evalf(add(1/ithprime(k)^5, k=1..i), 100)); end:
A085965(100000); # Paolo P. Lava, May 29 2012
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MATHEMATICA
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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-François Alcover, Feb 14 2013, from 1st formula *)
RealDigits[ PrimeZetaP[ 5], 10, 111][[1]] (* Robert G. Wilson v, Sep 03 2014 *)
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PROG
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(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
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CROSSREFS
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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
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KEYWORD
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cons,easy,nonn
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AUTHOR
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Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Jul 06 2003
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STATUS
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approved
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