A001899 Number of divisors of n of the form 5k+4; a(0) = 0.

0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 0, 0, 2, 0, 0, 1, 2, 1, 0, 0, 1, 0, 1, 0, 2, 0, 1, 1, 1, 0, 1, 0, 2, 1, 0, 0, 2, 1, 0, 0, 1, 0, 2, 0, 2, 1, 1, 1, 1, 0, 0, 1, 2, 0, 0, 0, 2, 1, 1, 0, 3, 0, 1, 0, 2, 0, 1, 1, 1, 1, 0, 0, 3, 0, 0

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Links

• T. D. Noe, Table of n, a(n) for n = 0..10000
• 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. A001876, A001877, A001878.

Cf. A001620, A256849.

Programs

Mathematica

Join[{0}, Table[d = Divisors[n]; Length[Select[d, Mod[#, 5] == 4 &]], {n, 100}]] (* T. D. Noe, Aug 10 2012 *)

PARI

a(n) = if (n==0, 0, sumdiv(n, d, (d % 5)==4)); \\ Michel Marcus, Feb 28 2021

Formula

G.f.: Sum_{n>=0} x^(5*n+4)/(1 - x^(5*n+4)).

G.f.: Sum_{k>=1} x^(4*k)/(1 - x^(5*k)). - Ilya Gutkovskiy, Sep 11 2019

Sum_{k=1..n} a(k) = n*log(n)/5 + c*n + O(n^(1/3)*log(n)), where c = gamma(4,5) - (1 - gamma)/5 = A256849 - (1 - A001620)/5 = -0.213442... (Smith and Subbarao, 1981). - Amiram Eldar, Nov 25 2023

Extensions

Better definition from Michael Somos, Aug 31 2004