A083913 Number of divisors of n that are congruent to 3 modulo 10.

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

Offset: 1

Author

Reinhard Zumkeller, May 08 2003

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. A000005, A001227, A010879.

Cf. A001620, A002392, A354642 (psi(3/10)).

Cf. A083910, A083911, A083912, A083914, A083915, A083916, A083917, A083918, A083919.

Programs

Mathematica

a[n_] := DivisorSum[n, 1 &, Mod[#, 10] == 3 &]; Array[a, 100] (* Amiram Eldar, Dec 30 2023 *)

PARI

a(n) = sumdiv(n, d, d % 10 == 3); \\ Amiram Eldar, Dec 30 2023

Formula

a(n) = A000005(n) - A083910(n) - A083911(n) - A083912(n) - A083914(n) - A083915(n) - A083916(n) - A083917(n) - A083918(n) - A083919(n).

G.f.: Sum_{k>=1} x^(3*k)/(1 - x^(10*k)). - Ilya Gutkovskiy, Sep 11 2019

Sum_{k=1..n} a(k) = n*log(n)/10 + c*n + O(n^(1/3)*log(n)), where c = gamma(3,10) - (1 - gamma)/10 = 0.07771547..., gamma(3,10) = -(psi(3/10) + log(10))/10 is a generalized Euler constant, and gamma is Euler's constant (A001620) (Smith and Subbarao, 1981). - Amiram Eldar, Dec 30 2023