A307869 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).
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, 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, 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
Offset: 1
Examples
1.42765653542483988311752393968732790409373362807443...
Links
- Ouarda Bouakkaz and Abdallah Derbal, The mean value of the function d(n)/d*(n) in arithmetic progressions, Notes on Number Theory and Discrete Mathematics, Vol. 29, No. 3 (2023), pp. 445-453.
- Mihoub Bouderbala, On a sum of a multiplicative function linked to the divisor function over the set of integers B-multiple of 5, Communications in Mathematics, Vol. 31, No. 1 (2023), pp. 369-374; arXiv preprint, arXiv:2212.05549 [math.NT], 2022.
- Abdallah Derbal and Meselem Karras, Valeurs moyennes d'une fonction liée aux diviseurs d'un nombre entier, C. R. Acad. Sci. Paris, Ser. I, Vol. 354, No. 6 (2016), pp. 555-558.
- Mark Kac, Statistical Independence in Probability, Analysis and Number Theory, Carus Monograph 12, Math. Assoc. Amer., 1959, p. 79.
Programs
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Mathematica
$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]] -
PARI
prodeulerrat(1 + 1/(2 * p * (p-1))) \\ Amiram Eldar, Mar 17 2021
Formula
Equals Product_{p prime} 1 + 1/(2 * p * (p-1)).
Equals (Pi^2/6) * Product_{p prime} 1 - 1/(2 * p^2) + 1/(2 * p^3).
Extensions
More terms from Vaclav Kotesovec, May 29 2020
Comments