A086242 Decimal expansion of the sum of 1/(p-1)^2 over all primes p.
1, 3, 7, 5, 0, 6, 4, 9, 9, 4, 7, 4, 8, 6, 3, 5, 2, 8, 7, 9, 1, 7, 2, 5, 3, 1, 3, 0, 5, 2, 2, 4, 3, 9, 6, 9, 9, 1, 7, 9, 5, 9, 9, 9, 6, 0, 1, 7, 5, 3, 1, 7, 4, 5, 8, 7, 0, 9, 1, 8, 9, 3, 3, 5, 8, 9, 1, 2, 3, 5, 7, 1, 3, 1, 4, 1, 5, 5, 5, 2, 5, 5, 4, 2, 9, 9, 0, 7, 6, 5, 2, 4, 1, 6, 5, 8, 8, 1, 1, 4, 5, 2, 7, 6, 0, 6, 5, 7, 4, 4, 8, 0, 6, 5, 7, 4
Offset: 1
Examples
1.37506499474863528791725313052243969917959996017...
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.
Links
- Henri Cohen, High precision computation of Hardy-Littlewood Constants (dvi), 1998.
- Henri Cohen, High-precision computation of Hardy-Littlewood constants. [pdf copy, with permission]
- Rafael Jakimczuk, On Sums of Powers of the p-adic Valuation of n!, Journal of Integer Sequences, Vol. 20 (2017), Article 17.5.6.
- Eric Weisstein's World of Mathematics, Prime Factor.
- Eric Weisstein's World of Mathematics, Prime Sums.
- Index to constants which are prime zeta sums {0,2,0}
Programs
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Mathematica
digits = 116; Np = NSum[(n-1)*PrimeZetaP[n], {n, 2, Infinity}, NSumTerms -> 3*digits, WorkingPrecision -> digits+10]; RealDigits[Np, 10, digits] // First (* Jean-François Alcover, Sep 02 2015 *) -
PARI
default(realprecision,256); (f(k)=return(sum(n=1,1024,moebius(n)/n*log(zeta(k*n))))); sum(k=2,1024,(k-1)*f(k)) /* Robert Gerbicz, Sep 12 2012 */
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PARI
sumeulerrat(1/(p-1)^2) \\ Amiram Eldar, Mar 19 2021
Formula
Equals Sum_{k>=2} (k-1)*primezeta(k). - Robert Gerbicz, Sep 12 2012
Equals Sum_{k>=2} (J_2(k)-phi(k)) * log(zeta(k)) / k, where J_2 = A007434 and phi = A000010 (Jakimczuk, 2017). - Amiram Eldar, Mar 18 2024
Extensions
More digits copied from Cohen's paper by R. J. Mathar, Dec 05 2008
More terms from Robert Gerbicz, Sep 12 2012