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A083916 Number of divisors of n that are congruent to 6 modulo 10.

%I A083916 #14 Dec 30 2023 04:12:03
%S A083916 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,
%T A083916 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,
%U A083916 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
%N A083916 Number of divisors of n that are congruent to 6 modulo 10.
%H A083916 Amiram Eldar, <a href="/A083916/b083916.txt">Table of n, a(n) for n = 1..10000</a>
%H A083916 R. A. Smith and M. V. Subbarao, <a href="https://doi.org/10.4153/CMB-1981-005-3">The average number of divisors in an arithmetic progression</a>, Canadian Mathematical Bulletin, Vol. 24, No. 1 (1981), pp. 37-41.
%F A083916 a(n) = A000005(n) - A083910(n) - A083911(n) - A083912(n) - A083913(n) - A083914(n) - A083915(n) - A083917(n) - A083918(n) - A083919(n).
%F A083916 G.f.: Sum_{k>=1} x^(6*k)/(1 - x^(10*k)). - _Ilya Gutkovskiy_, Sep 11 2019
%F A083916 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
%t A083916 Table[Count[Divisors[n],_?(Mod[#,10]==6&)],{n,110}] (* _Harvey P. Dale_, Oct 16 2013 *)
%t A083916 a[n_] := DivisorSum[n, 1 &, Mod[#, 10] == 6 &]; Array[a, 100] (* _Amiram Eldar_, Dec 30 2023 *)
%o A083916 (PARI) a(n) = sumdiv(n, d, d % 10 == 6); \\ _Amiram Eldar_, Dec 30 2023
%Y A083916 Cf. A000005, A001227, A010879.
%Y A083916 Cf. A001620, A002392, A200137 (psi(3/5)).
%Y A083916 Cf. A083910, A083911, A083912, A083913, A083914, A083915, A083917, A083918, A083919.
%K A083916 nonn,easy
%O A083916 1,36
%A A083916 _Reinhard Zumkeller_, May 08 2003