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%I #48 Mar 18 2024 05:29:41
%S 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,
%T 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,
%U 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
%N Decimal expansion of the sum of 1/(p-1)^2 over all primes p.
%D Henri Cohen, Number Theory, Volume II: Analytic and Modern Tools, GTM Vol. 240, Springer, 2007; see pp. 208-209.
%D Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, Vol. 94, Cambridge University Press, 2003, pp. 94-98.
%H Henri Cohen, <a href="http://www.math.u-bordeaux.fr/~cohen/hardylw.dvi">High precision computation of Hardy-Littlewood Constants (dvi)</a>, 1998.
%H Henri Cohen, <a href="/A221712/a221712.pdf">High-precision computation of Hardy-Littlewood constants</a>. [pdf copy, with permission]
%H Rafael Jakimczuk, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL20/Jakimczuk/jak37.html">On Sums of Powers of the p-adic Valuation of n!</a>, Journal of Integer Sequences, Vol. 20 (2017), Article 17.5.6.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimeFactor.html">Prime Factor</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimeSums.html">Prime Sums</a>.
%F Equals Sum_{k>=2} (k-1)*primezeta(k). - _Robert Gerbicz_, Sep 12 2012
%F Equals lim_{n -> oo} A119686(n)/A334746(n). - _Petros Hadjicostas_, May 11 2020
%F 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
%e 1.37506499474863528791725313052243969917959996017...
%t 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 *)
%o (PARI) default(realprecision,256);
%o (f(k)=return(sum(n=1,1024,moebius(n)/n*log(zeta(k*n)))));
%o sum(k=2,1024,(k-1)*f(k)) /* _Robert Gerbicz_, Sep 12 2012 */
%o (PARI) sumeulerrat(1/(p-1)^2) \\ _Amiram Eldar_, Mar 19 2021
%Y Cf. A000010, A007434, A119686, A334746.
%K nonn,cons
%O 1,2
%A _Eric W. Weisstein_, Jul 13 2003
%E More digits copied from Cohen's paper by _R. J. Mathar_, Dec 05 2008
%E More terms from _Robert Gerbicz_, Sep 12 2012
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