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A085548 Decimal expansion of the prime zeta function at 2: Sum_{p prime>=2} 1/p^2. 27
4, 5, 2, 2, 4, 7, 4, 2, 0, 0, 4, 1, 0, 6, 5, 4, 9, 8, 5, 0, 6, 5, 4, 3, 3, 6, 4, 8, 3, 2, 2, 4, 7, 9, 3, 4, 1, 7, 3, 2, 3, 1, 3, 4, 3, 2, 3, 9, 8, 9, 2, 4, 2, 1, 7, 3, 6, 4, 1, 8, 9, 3, 0, 3, 5, 1, 1, 6, 5, 0, 2, 7, 3, 6, 3, 9, 1, 0, 8, 7, 4, 4, 4, 8, 9, 5, 7, 5, 4, 4, 3, 5, 4, 9, 0, 6, 8, 5, 8, 2, 2, 2, 8, 0, 6 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

0,1

REFERENCES

Diaconis, Persi; Mosteller, Frederick; Onishi, Hironari. Second-order terms for the variances and covariances of the number of prime factors-including the square free case. J. Number Theory 9 (1977), no. 2, 187--202. MR0434991 (55 #7953).

S. R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, pp. 94-98

LINKS

Table of n, a(n) for n=0..104.

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

X. Gourdon and P. Sebah, Some Constants from Number theory

Gerhard Niklasch and Pieter Moree, Some number-theoretical constants [Cached copy]

Eric Weisstein's World of Mathematics, Prime Zeta Function

Eric Weisstein's World of Mathematics, Distinct Prime Factors

Wikipedia, Prime Zeta Function

FORMULA

P(2) = Sum_{p prime>=2} 1/p^2 = Sum_{n=1..inf} mobius(n)*log(zeta(2*n))/n. - Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Jul 06 2003

Equals A085991 + A086032 + 1/4. - R. J. Mathar, Jul 22 2010

EXAMPLE

0.4522474200410654985065... = 1/2^2 + 1/3^2 + 1/5^2 +1/7^2 + 1/11^2 + 1/13^2+...

MAPLE

A085548:= proc(i) print(evalf(add(1/ithprime(k)^2, k=1..i), 100)); end:

A085548(100000); Paolo P. Lava, May 29 2012

MATHEMATICA

If [$VersionNumber < 7.0, m = 200; $MaxExtraPrecision = 200; PrimeZetaP[s_] := NSum[MoebiusMu[k]*Log[Zeta[k*s]]/k, {k, 1, m}, AccuracyGoal -> m, NSumTerms -> m, PrecisionGoal -> m, WorkingPrecision -> m]]; RealDigits[PrimeZetaP[2]][[1]][[1 ;; 105]] (* Jean-Fran├žois Alcover, Jun 24 2011 *)

PROG

(PARI) recip2(n) = { v=0; p=1; forprime(y=2, n, v=v+1./y^2; ); print(v) }

(PARI) eps()=my(p=default(realprecision)); precision(2.>>(32*ceil(p*38539962/371253907)), 9)

lm=lambertw(log(4)/eps())\log(4);

sum(k=1, lm, moebius(k)/k*log(abs(zeta(2*k)))) \\ Charles R Greathouse IV, Jul 19 2013

CROSSREFS

Cf. A085541 (at 3), A136271 (derivative), A117543 (semiprimes), A222056, A209329, A124012.

Sequence in context: A156890 A163531 A016715 * A074459 A155793 A070593

Adjacent sequences:  A085545 A085546 A085547 * A085549 A085550 A085551

KEYWORD

easy,nonn,cons

AUTHOR

Cino Hilliard, Jul 03 2003

EXTENSIONS

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

Offset corrected by R. J. Mathar, Feb 05 2009

STATUS

approved

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Last modified April 23 04:16 EDT 2014. Contains 240909 sequences.