A336908 Decimal expansion of Sum_{p prime} (p^2 + p - 1)/(p^2 *(p - 1)^2).

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

Offset: 1

Author

Amiram Eldar, Aug 07 2020

Keywords

• nonn
• cons

Comments

The asymptotic variance of Omega(k) - omega(k) (A046660).

The asymptotic mean of Omega(k) - omega(k) is Sum_{p prime} 1/(p*(p-1)) = 0.773156... (A136141).

Examples

			1.695974243757364917275077225546134160625109953018611...
		

Links

• Persi Diaconis, Frederick Mosteller and Hironari Onishi, Second-order terms for the variances and covariances of the number of prime factors - Including the square free case, Journal of Number Theory, Vol. 9, No. 2 (1977), pp. 187-202.
• Ali Rejali, On the Asymptotic Expansions for the Moments and the Limiting Distributions of Some Additive Arithmetic Functions, Ph.D. dissertation, Department of Statistics, Stanford University, 1978. See p. 59.
• Eric Weisstein's World of Mathematics, Prime Zeta Function.
• Wikipedia, Prime zeta function.

Crossrefs

Cf. A001221, A001222, A046660, A136141.

Programs

Mathematica

m = 100; RealDigits[PrimeZetaP[2] + NSum[n * PrimeZetaP[n], {n, 3, Infinity}, WorkingPrecision -> 2*m, NSumTerms -> 3*m], 10, m][[1]]

PARI

sumeulerrat((p^2 + p - 1)/(p^2 *(p - 1)^2)) \\ Hugo Pfoertner, Aug 08 2020

Formula

Equals lim_{m->oo} (1/m) * Sum_{k=1..m} d(k)^2 - ((1/m) * Sum_{k=1..m} d(k))^2, where d(k) = Omega(k) - omega(k) = A001222(k) - A001221(k) = A046660(k).

Equals P(2) + Sum_{k>=3} k*P(k), where P is the prime zeta function.

Equals A086242 -A085548 +A136141 . - R. J. Mathar, Aug 19 2022