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

1,12

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.

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

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 *)

Prog

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

Crossrefs

Cf. A000005, A001227, A010879.

Cf. A001620, A002392, A200135 (psi(1/5)).

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

Sequence in context: A216188 A117145 A338822 * A256003 A157187 A152140

Adjacent sequences: A083909 A083910 A083911 * A083913 A083914 A083915

Keyword

nonn,easy

Author

Reinhard Zumkeller, May 08 2003

Status

approved