A363795 Number of divisors of n of the form 7*k + 2.

0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 2, 0, 2, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 2, 0, 2, 0, 1, 0, 2, 1, 1, 0, 1, 0, 1, 0, 2, 1, 2, 0, 2, 0, 1, 1, 1, 0, 2, 0, 1, 0, 2, 0, 2, 0, 1, 1, 2, 1, 1, 0, 1, 1, 1, 0, 3, 0, 2, 0, 1, 0, 1, 1, 2, 1, 1, 0, 1, 0, 2, 0, 2, 0, 3, 0, 2, 1, 1, 0, 2, 0, 1, 1

Offset: 1

Author

Seiichi Manyama, Jun 23 2023

Keywords

• nonn
• easy

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.

Crossrefs

Cf. A279061, A363805, A363806, A363807, A363808.

Cf. A001822, A001877.

Cf. A284443.

Cf. A001620, A016630, A354628 (psi(2/7)).

Programs

Mathematica

a[n_] := DivisorSum[n, 1 &, Mod[#, 7] == 2 &]; Array[a, 100] (* Amiram Eldar, Jun 23 2023 *)

PARI

a(n) = sumdiv(n, d, d%7==2);

Formula

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

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

Sum_{k=1..n} a(k) = n*log(n)/7 + c*n + O(n^(1/3)*log(n)), where c = gamma(2,7) - (1 - gamma)/7 = 0.188117..., gamma(2,7) = -(psi(2/7) + log(7))/7 is a generalized Euler constant, and gamma is Euler's constant (A001620) (Smith and Subbarao, 1981). - Amiram Eldar, Nov 25 2023