A188172 Number of divisors d of n of the form d == 7 (mod 8).

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

Offset: 1

Author

R. J. Mathar, Mar 23 2011

Keywords

• nonn
• easy

Examples

			a(A007522(i)) = 1, any i.
		

Links

• Reinhard Zumkeller, 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, A004771, A141164, A188169, A188170, A188171, A188226.

Cf. A001620, A016631, A354635 (psi(7/8)).

Programs

Haskell

a188172 n = length $ filter ((== 0) . mod n) [7,15..n]
-- Reinhard Zumkeller, Mar 26 2011

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:
A188172 := proc(n) sigmamr(n,8,7) ; end proc:

Mathematica

Table[Count[Divisors[n],?(Mod[#,8]==7&)],{n,90}] (* _Harvey P. Dale, Mar 08 2014 *)

PARI

a(n) = sumdiv(n, d, (d % 8) == 7); \\ Amiram Eldar, Nov 25 2023

Formula

A188170(n)+a(n) = A001842(n).

A188169(n)+A188170(n)-A188171(n)-a(n) = A002325(n).

a(A188226(n))=n and a(m)<>n for m<A188226(n), n>=0; a(A141164(n))=1. - Reinhard Zumkeller, Mar 26 2011

G.f.: Sum_{k>=1} x^(7*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(7,8) - (1 - gamma)/8 = -0.212276..., gamma(7,8) = -(psi(7/8) + log(8))/8 is a generalized Euler constant, and gamma is Euler's constant (A001620) (Smith and Subbarao, 1981). - Amiram Eldar, Nov 25 2023