A083912 Number of divisors of n that are congruent to 2 modulo 10.

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

Offset: 1

Author

Reinhard Zumkeller, May 08 2003

Keywords

• nonn
• easy

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537
• 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, A200135 (psi(1/5)).

Cf. A083910, A083911, A083913, A083914, A083915, A083916, A083917, A083918, A083919.

Programs

Mathematica

a[n_] := Sum[If[Mod[d, 10] == 2, 1, 0], {d, Divisors[n]}];
Array[a, 105] (* Jean-François Alcover, Dec 02 2021 *)
a[n_] := DivisorSum[n, 1 &, Mod[#, 10] == 2 &]; Array[a, 100] (* Amiram Eldar, Dec 30 2023 *)

PARI

A083912(n) = sumdiv(n,d,2==(d%10)); \\ Antti Karttunen, Jan 22 2020

Formula

a(n) = A000005(n) - A083910(n) - A083911(n) - A083913(n) - A083914(n) - A083915(n) - A083916(n) - A083917(n) - A083918(n) - A083919(n).

G.f.: Sum_{k>=1} x^(2*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(2,10) - (1 - gamma)/10 = 0.256367..., gamma(2,10) = -(psi(1/5) + log(10))/10 is a generalized Euler constant, and gamma is Euler's constant (A001620) (Smith and Subbarao, 1981). - Amiram Eldar, Dec 30 2023