A001878 Number of divisors of n of the form 5k+3; a(0) = 0.
0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 2, 0, 0, 1, 0, 1, 2, 0, 1, 1, 1, 0, 1, 0, 1, 2, 0, 0, 2, 0, 1, 2, 1, 0, 1, 1, 0, 1, 1, 0, 3, 0, 0, 1, 1, 1, 2, 0, 2, 1, 1, 0, 1, 0, 0, 2, 1, 1, 2, 0, 1, 2, 0, 0, 3, 1, 0, 1, 1, 0, 3, 0, 1, 1, 0, 1, 2, 0, 1, 1
Offset: 0
Links
- T. D. Noe, Table of n, a(n) for n = 0..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.
Programs
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Mathematica
Join[{0}, Table[d = Divisors[n]; Length[Select[d, Mod[#, 5] == 3 &]], {n, 100}]] (* T. D. Noe, Aug 10 2012 *) Table[Count[Divisors[n],?(Mod[#,5]==3&)],{n,0,90}] (* _Harvey P. Dale, Nov 08 2012 *) -
PARI
a(n) = if (n==0, 0, sumdiv(n, d, (d % 5)==3)); \\ Michel Marcus, Feb 28 2021
Formula
G.f.: Sum_{n>=0} x^(5*n+3)/(1 - x^(5*n+3)).
G.f.: Sum_{k>=1} x^(3*k)/(1 - x^(5*k)). - Ilya Gutkovskiy, Sep 11 2019
Sum_{k=1..n} a(k) = n*log(n)/5 + c*n + O(n^(1/3)*log(n)), where c = gamma(3,5) - (1 - gamma)/5 = A256848 - (1 - A001620)/5 = -0.0983206... (Smith and Subbarao, 1981). - Amiram Eldar, Nov 25 2023