A319995 Number of divisors of n of the form 6*k + 5.

0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 2, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 0, 2, 0, 0, 1, 1, 1, 0, 0, 0, 0, 2, 1, 0, 1, 1, 2, 1, 0, 0, 0, 1, 0, 2, 0, 0, 1, 0, 1, 1, 0, 2, 0, 1, 1, 1, 1, 0, 1, 0, 1, 2, 0, 0, 0, 1, 1, 1, 1, 0, 0, 2

Offset: 1

Author

Antti Karttunen, Oct 03 2018

Keywords

• nonn
• easy

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537
• R. A. Smith and M. V. Subbarao, The average number of divisors in an arithmetic progression, Canadian Mathematical Bulletin, Vol. 24, No. 1 (1981), pp. 37-41.

Crossrefs

Cf. A035218, A279060, A320005.

Cf. A001620, A016629, A222458 (psi(5/6)).

Programs

Mathematica

a[n_] := DivisorSum[n, 1 &, Mod[#, 6] == 5 &]; Array[a, 100] (* Amiram Eldar, Nov 25 2023 *)

PARI

A319995(n) = if(!n,n,sumdiv(n, d, (5==(d%6))));

Formula

a(n) = A035218(n) - A279060(n).

G.f.: Sum_{k>=1} x^(5*k)/(1 - x^(6*k)). - Ilya Gutkovskiy, Sep 11 2019

Sum_{k=1..n} a(k) = n*log(n)/6 + c*n + O(n^(1/3)*log(n)), where c = gamma(5,6) - (1 - gamma)/6 = -0.220635..., gamma(5,6) = -(psi(5/6) + log(6))/6 is a generalized Euler constant, and gamma is Euler's constant (A001620) (Smith and Subbarao, 1981). - Amiram Eldar, Nov 25 2023