A218447 a(n) = Sum_{k>=0} floor(n/(5*k + 4)).

0, 0, 0, 0, 1, 1, 1, 1, 2, 3, 3, 3, 4, 4, 5, 5, 6, 6, 7, 8, 9, 9, 9, 9, 11, 11, 11, 12, 14, 15, 15, 15, 16, 16, 17, 17, 19, 19, 20, 21, 22, 22, 23, 23, 25, 26, 26, 26, 28, 29, 29, 29, 30, 30, 32, 32, 34, 35, 36, 37, 38, 38, 38, 39, 41, 41, 41, 41, 43, 44, 45, 45, 48, 48, 49, 49, 51, 51, 52, 53, 54

Offset: 0

Author

Benoit Cloitre, Oct 28 2012

Keywords

• nonn
• easy

Links

• Robert Israel, 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

Partial sums of A001899.

Cf. A218444, A218445, A218446.

Cf. A001620, A256849.

Programs

Maple

g:= n -> nops(select(t -> t mod 5 = 4, numtheory:-divisors(n))):
g(0):= 0:
ListTools:-PartialSums(map(g, [$0..100])); # Robert Israel, Apr 29 2021

Maxima

A218447[n]:=sum(floor(n/(5*k+4)),k,0,n)$
makelist(A218447[n],n,0,80); /* Martin Ettl, Oct 20 2012 */

PARI

a(n)=sum(k=0,n,(n\(5*k+4)))

Formula

a(n) = 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, Apr 20 2025