A363807 Number of divisors of n of the form 7*k + 5.

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

Offset

1,40

Links

Amiram Eldar, Table of n, a(n) for n = 1..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.

Formula

G.f.: Sum_{k>0} x^(5*k)/(1 - x^(7*k)).

G.f.: Sum_{k>0} x^(7*k-2)/(1 - x^(7*k-2)).

Sum_{k=1..n} a(k) = n*log(n)/7 + c*n + O(n^(1/3)*log(n)), where c = gamma(5,7) - (1 - gamma)/7 = -0.169787..., gamma(5,7) = -(psi(5/7) + log(7))/7 is a generalized Euler constant, and gamma is Euler's constant (A001620) (Smith and Subbarao, 1981). - Amiram Eldar, Nov 25 2023

Mathematica

a[n_] := DivisorSum[n, 1 &, Mod[#, 7] == 5 &]; Array[a, 100] (* Amiram Eldar, Jun 23 2023 *)

Prog

(PARI) a(n) = sumdiv(n, d, d%7==5);

Crossrefs

Cf. A279061, A363795, A363805, A363806, A363808.

Cf. A284446, A319995.

Cf. A001620, A016630, A354631 (psi(5/7)).

Sequence in context: A101668 A141846 A188171 * A330733 A328496 A035202

Adjacent sequences: A363804 A363805 A363806 * A363808 A363809 A363810

Keyword

nonn,easy

Author

Seiichi Manyama, Jun 23 2023

Status

approved