A136271 Decimal expansion of sum log(p)/p^2 over the primes p=2,3,5,7,11,...

4, 9, 3, 0, 9, 1, 1, 0, 9, 3, 6, 8, 7, 6, 4, 4, 6, 2, 1, 9, 7, 8, 2, 6, 2, 0, 5, 0, 5, 6, 4, 9, 1, 2, 5, 8, 0, 5, 5, 5, 8, 8, 1, 2, 6, 3, 4, 6, 4, 6, 8, 2, 9, 0, 7, 1, 3, 3, 2, 7, 1, 2, 0, 3, 2, 1, 3, 3, 6, 7, 7, 3, 6, 7, 9, 5, 7, 8, 5, 2, 0, 3, 5, 5, 0, 7, 6, 0, 0, 4, 2, 1, 8, 1, 6, 9, 3, 1, 1, 2, 4, 2, 4, 6

Offset: 0

Author

R. J. Mathar, Mar 09 2008

Keywords

• cons
• nonn

Comments

The negative first derivative of the prime zeta function at 2.

Examples

			0.493091109368764462197826205056491258055588126346468290713327120321336...
		

References

• Henri Cohen, Number Theory, Volume II: Analytic and Modern Tools, GTM Vol. 240, Springer, 2007; see pp. 208-209.

Links

• Henri Cohen, High-precision computation of Hardy-Littlewood constants, preprint 1998.
• Henri Cohen, High-precision computation of Hardy-Littlewood constants. [pdf copy, with permission]
• Tengiz O. Gogoberidze, Baker's dozen digits of two sums involving reciprocal products of an integer and its greatest prime factor, arXiv:2407.12047 [math.GM], 2024. See p. 2.
• R. J. Mathar, Series of reciprocal powers of k-almost primes, arXiv:0803.0900 [math.NT], Table 3.
• J. B. Rosser, L. Schoenfeld, Approximate formulas for some functions of prime numbers, Ill. J. Math. 6 (1) (1962) 64-94, Table IV

Crossrefs

Cf. A303493-A303499, A360094, A360095.

Programs

Mathematica

RealDigits[PrimeZetaP'[2], 10, 104] // First (* Jean-François Alcover, Sep 11 2015 *)

Formula

Equals Sum_{n >= 1} log(A000040(n))/A001248(n).

Extensions

Removed 6 digits where preprints disagree. Added zero to an A-number in formula. Corrected Cohen preprint year. Added 2nd link. - R. J. Mathar, Nov 27 2008

More digits from Jean-François Alcover, Sep 11 2015