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

1,2

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

Table of n, a(n) for n=1..116.

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.

Formula

Equals Sum_{k>=2} (k-1)*primezeta(k). - Robert Gerbicz, Sep 12 2012

Equals lim_{n -> oo} A119686(n)/A334746(n). - Petros Hadjicostas, May 11 2020

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

Example

1.37506499474863528791725313052243969917959996017...

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 *)

Prog

(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 */
(PARI) sumeulerrat(1/(p-1)^2) \\ Amiram Eldar, Mar 19 2021

Crossrefs

Cf. A000010, A007434, A119686, A334746.

Sequence in context: A301755 A302558 A193506 * A322931 A096627 A065084

Adjacent sequences: A086239 A086240 A086241 * A086243 A086244 A086245

Keyword

nonn,cons

Author

Eric W. Weisstein, Jul 13 2003

Extensions

More digits copied from Cohen's paper by R. J. Mathar, Dec 05 2008

More terms from Robert Gerbicz, Sep 12 2012

Status

approved