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%I #42 Aug 10 2023 03:28:47
%S 1,4,2,7,6,5,6,5,3,5,4,2,4,8,3,9,8,8,3,1,1,7,5,2,3,9,3,9,6,8,7,3,2,7,
%T 9,0,4,0,9,3,7,3,3,6,2,8,0,7,4,4,3,9,2,7,4,2,2,4,7,4,1,4,3,6,7,3,4,4,
%U 2,9,8,8,3,4,1,1,5,3,8,9,4,0,7,4,8,3,0,3,5,2,6,0,8,3,7,4,0,5,1,7,7,9,3,2,5
%N Decimal expansion of the asymptotic mean of d(k)/2^omega(k), where d(k) is the number of divisors of k (A000005) and omega(k) is the number of its distinct prime divisors (A001221).
%C Also the asymptotic mean of ratio between the number of divisors and the number of unitary divisors of the integers.
%H Ouarda Bouakkaz and Abdallah Derbal, <a href="https://doi.org/10.7546/nntdm.2023.29.3.445-453">The mean value of the function d(n)/d*(n) in arithmetic progressions</a>, Notes on Number Theory and Discrete Mathematics, Vol. 29, No. 3 (2023), pp. 445-453.
%H Mihoub Bouderbala, <a href="https://doi.org/10.46298/cm.10467">On a sum of a multiplicative function linked to the divisor function over the set of integers B-multiple of 5</a>, Communications in Mathematics, Vol. 31, No. 1 (2023), pp. 369-374; <a href="https://arxiv.org/abs/2212.05549v2">arXiv preprint</a>, arXiv:2212.05549 [math.NT], 2022.
%H Abdallah Derbal and Meselem Karras, <a href="https://doi.org/10.1016/j.crma.2016.03.007">Valeurs moyennes d'une fonction liée aux diviseurs d'un nombre entier</a>, C. R. Acad. Sci. Paris, Ser. I, Vol. 354, No. 6 (2016), pp. 555-558.
%H Mark Kac, <a href="https://web.archive.org/web/20220609160228/http://www.gibbs.if.usp.br/~marchett/estocastica/MarkKac-Statistical-Independence.pdf">Statistical Independence in Probability, Analysis and Number Theory</a>, Carus Monograph 12, Math. Assoc. Amer., 1959, p. 79.
%F Equals Product_{p prime} 1 + 1/(2 * p * (p-1)).
%F Equals (Pi^2/6) * Product_{p prime} 1 - 1/(2 * p^2) + 1/(2 * p^3).
%e 1.42765653542483988311752393968732790409373362807443...
%t $MaxExtraPrecision = 1000; m = 1000; c = LinearRecurrence[{4, -6, 4}, {0, 4, 12}, m]; RealDigits[Exp[NSum[Indexed[c, n]*PrimeZetaP[n]/n/2^n, {n, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 100][[1]]
%o (PARI) prodeulerrat(1 + 1/(2 * p * (p-1))) \\ _Amiram Eldar_, Mar 17 2021
%Y Cf. A000005, A001221, A034444, A307870.
%K nonn,cons
%O 1,2
%A _Amiram Eldar_, May 02 2019
%E More terms from _Vaclav Kotesovec_, May 29 2020
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