A083914 Number of divisors of n that are congruent to 4 modulo 10.

0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 2, 0, 0, 0, 2, 0, 0, 0, 1, 0, 1, 0, 2, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 2, 0, 1, 0, 2, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 3, 0, 0, 0, 2, 0, 0, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 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..10000
• Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000
• 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, A200136 (psi(2/5)).

Cf. A083910, A083911, A083912, A083913, A083915, A083916, A083917, A083918, A083919.

Programs

Mathematica

Table[Count[Divisors[n],?(Mod[#,10]==4&)],{n,110}] (* _Harvey P. Dale, Dec 09 2014 *)
a[n_] := DivisorSum[n, 1 &, Mod[#, 10] == 4 &]; Array[a, 100] (* Amiram Eldar, Dec 30 2023 *)

PARI

A083914(n) = sumdiv(n,d,(4==(d%10))); \\ Antti Karttunen, Nov 07 2018

Formula

a(n) = A000005(n) - A083910(n) - A083911(n) - A083912(n) - A083913(n) - A083915(n) - A083916(n) - A083917(n) - A083918(n) - A083919(n).

G.f.: Sum_{k>=1} x^(4*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(4,10) - (1 - gamma)/10 = -0.0163984..., gamma(4,10) = -(psi(2/5) + log(10))/10 is a generalized Euler constant, and gamma is Euler's constant (A001620) (Smith and Subbarao, 1981). - Amiram Eldar, Dec 30 2023