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

1,63

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.

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

Example

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

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 *)

Prog

(Haskell)
a188172 n = length $ filter ((== 0) . mod n) [7, 15..n]
-- Reinhard Zumkeller, Mar 26 2011
(PARI) a(n) = sumdiv(n, d, (d % 8) == 7); \\ Amiram Eldar, Nov 25 2023

Crossrefs

Cf. A001842, A002325, A004771, A141164, A188169, A188170, A188171, A188226.

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

Sequence in context: A083895 A093488 A085858 * A106671 A033776 A117371

Adjacent sequences: A188169 A188170 A188171 * A188173 A188174 A188175

Keyword

nonn,easy

Author

R. J. Mathar, Mar 23 2011

Status

approved