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A085969 Decimal expansion of the prime zeta function at 9. 18
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

-2,1

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

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 = -2..1999

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

Equals Sum_{k>=1} 1/A179665(k). - Amiram Eldar, Jul 27 2020

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

(PARI) sumeulerrat(1/p, 9) \\ Hugo Pfoertner, Feb 03 2020

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, A179665.

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

EXTENSIONS

Changed offset and adapted data by Hugo Pfoertner, Jan 31 2020

STATUS

approved

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Last modified December 1 22:24 EST 2020. Contains 338858 sequences. (Running on oeis4.)