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A085969 Decimal expansion of the prime zeta function at 9. 13
0, 0, 2, 0, 0, 4, 4, 6, 7, 5, 7, 4, 9, 6, 2, 4, 5, 0, 6, 6, 3, 0, 7, 3, 5, 8, 5, 1, 4, 0, 7, 8, 3, 1, 1, 7, 5, 3, 6, 8, 2, 2, 9, 2, 0, 3, 4, 9, 7, 3, 8, 5, 7, 8, 5, 5, 0, 2, 7, 5, 5, 9, 1, 7, 3, 9, 7, 9, 3, 7, 2, 6, 2, 5, 1, 5, 1, 9, 9, 7, 7, 1, 9, 0, 8, 1, 3, 5, 5, 3, 4, 3, 4, 0, 1, 2, 4, 4, 1, 7, 7, 2, 4, 3, 0 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Mathar's Table 1 (cited below) lists expansions of the prime zeta function at integers s in 10..39. - Jason Kimberley, Jan 07 2017

REFERENCES

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..2003

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

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(9) = Sum_{p prime>=2} 1/p^9 = Sum_{n=1..inf} mobius(n)*log(zeta(9*n))/n.

EXAMPLE

0.0020044675749624506630...

MAPLE

A085969:= proc(i) print(evalf(add(1/ithprime(k)^9, k=1..i), 100)); end:

A085969(100000); # Paolo P. Lava, May 29 2012

MATHEMATICA

pz9[n_] := pz9[n] = Join[{0, 0}, Sum[ MoebiusMu[k]*Log[Zeta[9*k]]/k, {k, 1, n}] // RealDigits[#, 10, 103]& // First]; pz9[100]; pz9[n = 200]; While[pz9[n] != pz9[n - 100], n = n + 100]; pz9[n] (* Jean-Fran├žois Alcover, Feb 14 2013, from formula *)

RealDigits[ PrimeZetaP[ 9], 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, 0] cat Reverse(IntegerToSequence(Floor(PrimeZeta(9, 38)*10^105)));

// Jason Kimberley, Dec 30 2016

CROSSREFS

Decimal expansion of the prime zeta function: A085548 (at 2), A085541 (at 3), A085964 (at 4) to A085968 (at 8), this sequence (at 9).

Cf. A013667.

Sequence in context: A137505 A107498 A094295 * A117434 A115179 A131742

Adjacent sequences:  A085966 A085967 A085968 * A085970 A085971 A085972

KEYWORD

cons,easy,nonn

AUTHOR

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

STATUS

approved

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Last modified December 13 14:58 EST 2017. Contains 295958 sequences.