A188171 The number of divisors d of n of the form d == 5 (mod 8).

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

Offset

1,45

Comments

a(5n) >= 1 as d=5 contributes to the count.

Links

Antti Karttunen, 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.

Formula

A188169(n)+a(n) = A001826(n).

A188169(n)+A188170(n)-a(n)-A188172(n) = A002325(n).

G.f.: Sum_{k>=1} x^(5*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(5,8) - (1 - gamma)/8 = -0.131189..., gamma(5,8) = -(psi(5/8) + log(8))/8 is a generalized Euler constant, and gamma is Euler's constant (A001620) (Smith and Subbarao, 1981). - Amiram Eldar, Nov 25 2023

Example

a(13) = 1 because the divisor d=13 is 8+5 == 5 (mod 8).

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:
A188171 := proc(n) sigmamr(n, 8, 5) ; end proc:

Mathematica

a[n_] := DivisorSum[n, 1 &, Mod[#, 8] == 5 &]; Array[a, 100] (* Amiram Eldar, Nov 25 2023 *)

Prog

(PARI) A188171(n) = sumdiv(n, d, (5==(d%8)));  \\ Antti Karttunen, Jul 09 2017

Crossrefs

Cf. A001826, A002325, A188169, A188170, A188172.

Cf. A001620, A016631, A354634 (psi(5/8)).

Sequence in context: A242830 A101668 A141846 * A363807 A330733 A328496

Adjacent sequences: A188168 A188169 A188170 * A188172 A188173 A188174

Keyword

nonn,easy

Author

R. J. Mathar, Mar 23 2011

Status

approved