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A085548 Decimal expansion of the prime zeta function at 2: Sum_{p prime} 1/p^2. 88
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
COMMENTS
Mathar's Table 1 (cited below) lists expansions of the prime zeta function at integers s in 10..39. - Jason Kimberley, Jan 05 2017
REFERENCES
Henri Cohen, Number Theory, Volume II: Analytic and Modern Tools, GTM Vol. 240, Springer, 2007; see pp. 208-209.
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, pp. 94-98.
J. W. L. Glaisher, On the Sums of Inverse Powers of the Prime Numbers, Quart. J. Math. 25, 347-362, 1891.
LINKS
Henri Cohen, High-precision computation of Hardy-Littlewood constants. [pdf copy, with permission]
Persi Diaconis, Frederick Mosteller, and 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).
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 171 and 190.
Xavier Gourdon  and  Pascal Sebah, Some Constants from Number theory.
Shanta Laishram and Florian Luca, Rectangles Of Nonvisible Lattice Points, J. Int. Seq. 18 (2015), Article 15.10.8, Theorem 1.
Jon Lee, Joseph Paat, Ingo Stallknecht, and Luze Xu, Polynomial upper bounds on the number of differing columns of Delta-modular integer programs, arXiv:2105.08160 [math.OC], 2021, see page 23.
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, Distinct Prime Factors.
Eric Weisstein's World of Mathematics, Prime Sums.
Eric Weisstein's World of Mathematics, Prime Zeta Function.
FORMULA
P(2) = Sum_{p prime} 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
Equals Sum_{k>=1} 1/A001248(k). - Amiram Eldar, Jul 27 2020
Equals Sum_{k>=2} pi(k)*(2*k+1)/(k^2*(k+1)^2), where pi(k) = A000720(k) (Shamos, 2011, p. 9). - Amiram Eldar, Mar 12 2024
EXAMPLE
0.4522474200410654985065... = 1/2^2 + 1/3^2 + 1/5^2 +1/7^2 + 1/11^2 + 1/13^2 + ...
MATHEMATICA
RealDigits[PrimeZetaP[2], 10, 105][[1]] (* Jean-François Alcover, Jun 24 2011, updated May 06 2021 *)
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
(PARI) sumeulerrat(1/p, 2) \\ Hugo Pfoertner, Feb 03 2020
(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
CROSSREFS
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.
Sequence in context: A016715 A337192 A255701 * A329957 A074459 A371747
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 20 06:37 EDT 2024. Contains 371799 sequences. (Running on oeis4.)