A083916 Number of divisors of n that are congruent to 6 modulo 10.
0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 2, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 3, 0, 0, 0, 0, 0, 1, 0, 1, 0
Offset: 1
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
Programs
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Mathematica
Table[Count[Divisors[n],?(Mod[#,10]==6&)],{n,110}] (* _Harvey P. Dale, Oct 16 2013 *) a[n_] := DivisorSum[n, 1 &, Mod[#, 10] == 6 &]; Array[a, 100] (* Amiram Eldar, Dec 30 2023 *)
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PARI
a(n) = sumdiv(n, d, d % 10 == 6); \\ Amiram Eldar, Dec 30 2023
Formula
a(n) = A000005(n) - A083910(n) - A083911(n) - A083912(n) - A083913(n) - A083914(n) - A083915(n) - A083917(n) - A083918(n) - A083919(n).
G.f.: Sum_{k>=1} x^(6*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(6,10) - (1 - gamma)/10 = -0.118475..., gamma(6,10) = -(psi(3/5) + log(10))/10 is a generalized Euler constant, and gamma is Euler's constant (A001620) (Smith and Subbarao, 1981). - Amiram Eldar, Dec 30 2023