A001876 Number of divisors of n of the form 5k+1; a(0)=0.

0, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 2, 1, 1, 2, 2, 1, 2, 1, 2, 1, 1, 1, 2, 2, 2, 2, 1, 1, 3, 1, 1, 1, 1, 2, 3, 1, 2, 1, 2, 1, 3, 1, 1, 2, 2, 1, 2, 2, 2, 1, 1, 1, 2, 2, 2, 2, 2, 1, 4, 1, 1, 1, 1, 2, 3, 1, 1, 1, 2, 2, 3, 1, 2, 2, 2, 1, 3, 1, 2, 1

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

For numbers of divisors of n of the form 5k+i (i=1, 2, 3, 4) see: this sequence, A001877, A001878, A001899.

Cf. A001620, A256779.

Programs

Mathematica

CoefficientList[ Series[ Together[ Sum[ x^n/(1 - x^(5n)), {n, 110}]], {x, 0, 110}], x] (* Robert G. Wilson v, Jan 31 2011 *)
a[n_] := DivisorSum[n, 1 &, Mod[#, 5] == 1 &]; a[0] = 0; Array[a, 100, 0] (* Amiram Eldar, Nov 25 2023 *)

PARI

a(n) = if(n==0,0, sumdiv(n, d, (d % 5) == 1)); \\ Michel Marcus, Feb 25 2021

Formula

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

G.f.: Sum_{n>=1} x^n/(1-x^(5*n)). - Joerg Arndt, Jan 30 2011

Sum_{k=1..n} a(k) = n*log(n)/5 + c*n + O(n^(1/3)*log(n)), where c = gamma(1,5) - (1 - gamma)/5 = A256779 - (1 - A001620)/5 = 0.651363... (Smith and Subbarao, 1981). - Amiram Eldar, Nov 25 2023