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A336908
Decimal expansion of Sum_{p prime} (p^2 + p - 1)/(p^2 *(p - 1)^2).
0
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
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.
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
KEYWORD
nonn,cons
AUTHOR
Amiram Eldar, Aug 07 2020
STATUS
approved