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

1,2

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).

Links

Table of n, a(n) for n=1..87.

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.

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

Example

1.695974243757364917275077225546134160625109953018611...

Mathematica

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

Prog

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

Crossrefs

Cf. A001221, A001222, A046660, A136141.

Sequence in context: A335204 A121821 A021859 * A330594 A371534 A196462

Adjacent sequences: A336905 A336906 A336907 * A336909 A336910 A336911

Keyword

nonn,cons

Author

Amiram Eldar, Aug 07 2020

Status

approved