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A085548
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Decimal expansion of the prime zeta function at 2: Sum_{p prime>=2} 1/p^2.
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35
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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
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OFFSET
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0,1
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COMMENTS
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Mathar's Table 1 (cited below) lists expansions of the prime zeta function at integers s in 10..39. - Jason Kimberley, Jan 05 2017
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REFERENCES
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S. R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, pp. 94-98.
J. W. L. Glaisher, On the Sums of Inverse Powers of the Prime Numbers, Quart. J. Math. 25, 347-362, 1891.
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LINKS
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Jason Kimberley, Table of n, a(n) for n = 0..1093
H. Cohen, High Precision Computation of Hardy-Littlewood Constants, Preprint.
Persi Diaconis, Frederick Mosteller, Hironari Onishi, 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).
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.
Gerhard Niklasch and Pieter Moree, Some number-theoretical constants [Cached copy]
Eric Weisstein's World of Mathematics, Prime Sums
Eric Weisstein's World of Mathematics, Prime Zeta Function
Eric Weisstein's World of Mathematics, Distinct Prime Factors
Wikipedia, Prime Zeta Function
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FORMULA
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P(2) = Sum_{p prime>=2} 1/p^2 = Sum_{n>=1} 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
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EXAMPLE
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0.4522474200410654985065... = 1/2^2 + 1/3^2 + 1/5^2 +1/7^2 + 1/11^2 + 1/13^2 + ...
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MAPLE
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A085548:= proc(i) print(evalf(add(1/ithprime(k)^2, k=1..i), 100)); end:
A085548(100000); # Paolo P. Lava, May 29 2012
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MATHEMATICA
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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-François Alcover, Jun 24 2011 *)
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PROG
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(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
(MAGMA) R := RealField(106);
PrimeZeta := func<k, N |&+[R|MoebiusMu(n)/n*Log(ZetaFunction(R, k*n)):n in[1..N]]>;
Reverse(IntegerToSequence(Floor(PrimeZeta(2, 173)*10^105)));
// Jason Kimberley, Dec 30 2016
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CROSSREFS
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Decimal expansion of the prime zeta function: this sequence (at 2), A085541 (at 3), A085964 (at 4) to A085969 (at 9).
Cf. A136271 (derivative), A117543 (semiprimes), A222056, A209329, A124012.
Cf. A013661, A078437, A242301.
Sequence in context: A267095 A016715 A255701 * A074459 A155793 A070593
Adjacent sequences: A085545 A085546 A085547 * A085549 A085550 A085551
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KEYWORD
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easy,nonn,cons
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AUTHOR
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Cino Hilliard, Jul 03 2003
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EXTENSIONS
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More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Jul 06 2003
Offset corrected by R. J. Mathar, Feb 05 2009
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STATUS
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approved
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