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A085966 Decimal expansion of the prime zeta function at 6. 7
0, 1, 7, 0, 7, 0, 0, 8, 6, 8, 5, 0, 6, 3, 6, 5, 1, 2, 9, 5, 4, 1, 3, 3, 6, 7, 3, 2, 6, 6, 0, 5, 9, 3, 9, 9, 2, 0, 9, 5, 8, 5, 9, 4, 1, 8, 7, 4, 5, 4, 4, 2, 4, 4, 7, 3, 3, 1, 6, 3, 3, 6, 8, 8, 3, 6, 9, 6, 9, 7, 3, 6, 7, 4, 7, 1, 7, 2, 4, 3, 6, 6, 7, 1, 8, 6, 0, 3, 5, 0, 0, 7, 8, 1, 8, 0, 6, 2, 3, 0, 2, 8, 8, 2, 3 (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..1802

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(6) = Sum_{p prime>=2} 1/p^6 = Sum_{n>=1} mobius(n)*log(zeta(6*n))/n

Equals 1/2^6 + A085995 + A086036. - R. J. Mathar, Jul 14 2012

EXAMPLE

0.0170700868506365129541...

MAPLE

A085966:= proc(i) print(evalf(add(1/ithprime(k)^6, k=1..i), 100)); end:

A085966(100000); # Paolo P. Lava, May 29 2012

MATHEMATICA

s[n_] := s[n] = Sum[ MoebiusMu[k]*Log[Zeta[6*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 *)

RealDigits[ PrimeZetaP[ 6], 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(6, 57)*10^105)));

// Jason Kimberley, Dec 30 2016

CROSSREFS

Decimal expansion of the prime zeta function: A085548 (at 2), A085541 (at 3), A085964 (at 4), A085965 (at 5), this sequence (at 6), A085967 (at 7) to A085969 (at 9).

Cf. A013664.

Sequence in context: A225457 A188928 A036479 * A010678 A010503 A158857

Adjacent sequences:  A085963 A085964 A085965 * A085967 A085968 A085969

KEYWORD

cons,easy,nonn

AUTHOR

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

STATUS

approved

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Last modified January 23 01:55 EST 2018. Contains 298093 sequences.