A336908 - OEIS

%I #16 Aug 19 2022 05:39:10

%S 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,

%T 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,

%U 7,9,7,1,4,2,6,6,8,7,7,7,7,8,8,1,4,7,4

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

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

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

%H Persi Diaconis, Frederick Mosteller and Hironari Onishi, <a href="https://doi.org/10.1016/0022-314X(77)90022-1">Second-order terms for the variances and covariances of the number of prime factors - Including the square free case</a>, Journal of Number Theory, Vol. 9, No. 2 (1977), pp. 187-202.

%H Ali Rejali, <a href="https://purl.stanford.edu/pw192vg4427">On the Asymptotic Expansions for the Moments and the Limiting Distributions of Some Additive Arithmetic Functions</a>, Ph.D. dissertation, Department of Statistics, Stanford University, 1978. See p. 59.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimeZetaFunction.html">Prime Zeta Function</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Prime_zeta_function">Prime zeta function</a>.

%F 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).

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

%F Equals A086242 -A085548 +A136141 . - _R. J. Mathar_, Aug 19 2022

%e 1.695974243757364917275077225546134160625109953018611...

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

%o (PARI) sumeulerrat((p^2 + p - 1)/(p^2 *(p - 1)^2)) \\ _Hugo Pfoertner_, Aug 08 2020

%Y Cf. A001221, A001222, A046660, A136141.

%K nonn,cons

%O 1,2

%A _Amiram Eldar_, Aug 07 2020