A137250 Decimal expansion of the constant sum 1/(q*log(q)), summed over prime powers q > 1.

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

Offset: 1

Author

R. J. Mathar, Mar 09 2008

Keywords

• nonn
• cons

Comments

Evaluated from Sum_{m,k >= 1} A008683(k)*I(k*m)/k^2, where I(x) = Integral_{t=x..infinity} log zeta(t) dt is Cohen's underivative.

Examples

			2.0066664528310687...
		

References

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

Links

• D. A. Clark, An upper bound of sum 1/(a_i log a_i) for quasi-primitive sequences, Comp. Math. Appl., 35 (1998), 105-109.
• H. Cohen, High precision computation of Hardy-Littlewood constants, preprint, 1998.
• Henri Cohen, High-precision computation of Hardy-Littlewood constants. [pdf copy, with permission]
• R. J. Mathar, Twenty digits of some Integrals of the Prime Zeta Function, arXiv:0811.4739 [math.NT].

Crossrefs

Cf. A137245, A221711.

Programs

PARI

default(realprecision, 200); su = 0; for(s=1, 400, su = su + sum(k=1, 500, moebius(k)/k^2 * intnum(x=s*k,[[1], 1], log(zeta(x))))/s; print(su)); \\ Vaclav Kotesovec, Jun 12 2022

Formula

Equals Sum_{n>=2} 1/(A000961(n)*log(A000961(n))).

Equals Sum_{p primes} -log(1-1/p)/log(p). - Vaclav Kotesovec, Jun 12 2022

Extensions

8 more digits from R. J. Mathar, Dec 04 2008

More terms from Vaclav Kotesovec, Jun 12 2022