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

1,2

Comments

Also the asymptotic mean of ratio between the number of divisors and the number of unitary divisors of the integers.

Links

Table of n, a(n) for n=1..105.

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.

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).

Example

1.42765653542483988311752393968732790409373362807443...

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]]

Prog

(PARI) prodeulerrat(1 + 1/(2 * p * (p-1))) \\ Amiram Eldar, Mar 17 2021

Crossrefs

Cf. A000005, A001221, A034444, A307870.

Sequence in context: A260593 A143370 A367832 * A016695 A125271 A245262

Adjacent sequences: A307866 A307867 A307868 * A307870 A307871 A307872

Keyword

nonn,cons

Author

Amiram Eldar, May 02 2019

Extensions

More terms from Vaclav Kotesovec, May 29 2020

Status

approved