A188170 The number of divisors d of n of the form d == 3 (mod 8).

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

Offset: 1

Author

R. J. Mathar, Mar 23 2011

Keywords

• nonn
• easy

Comments

a(3n) >= 1 as the divisor d=3 contributes to the count then.

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000
• Michael D. Hirschhorn, The number of representations of a number by various forms, Discrete Mathematics 298 (2005), 205-211.
• 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. A001842, A002325, A188169, A188171, A188172.

Cf. A001620, A256782.

Programs

Maple

sigmamr := proc(n,m,r) local a,d ; a := 0 ; for d in numtheory[divisors](n) do if modp(d,m) = r then a := a+1 ; end if; end do: a; end proc:
A188170 := proc(n) sigmamr(n,8,3) ; end proc:

Mathematica

Table[Count[Divisors[n],?(Mod[#,8]==3&)],{n,100}] (* _Harvey P. Dale, Jul 08 2013 *)

PARI

a(n) = sumdiv(n, d, (d%8) == 3); \\ Michel Marcus, Nov 05 2018

Formula

a(n) + A188172(n) = A001842(n).

A188169(n) + a(n) - A188171(n) - A188172(n) = A002325(n).

G.f.: Sum_{k>=1} x^(3*k)/(1 - x^(8*k)). - Ilya Gutkovskiy, Sep 11 2019

Sum_{k=1..n} a(k) = n*log(n)/8 + c*n + O(n^(1/3)*log(n)), where c = gamma(3,8) - (1 - gamma)/8 = A256782 - (1 - A001620)/8 = 0.0314716... (Smith and Subbarao, 1981). - Amiram Eldar, Nov 25 2023