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A085966 Decimal expansion of the prime zeta function at 6. 13
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

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..1802

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(6) = Sum_{p prime} 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

Equals Sum_{k>=1} 1/A030516(k). - Amiram Eldar, Jul 27 2020

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

(PARI) sumeulerrat(1/p, 6) \\ Hugo Pfoertner, Feb 03 2020

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, A030516.

Sequence in context: A225457 A188928 A036479 * A010678 A010503 A335727

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 September 23 08:18 EDT 2020. Contains 337295 sequences. (Running on oeis4.)