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A083917 Number of divisors of n that are congruent to 7 modulo 10.

%I A083917 #14 Dec 30 2023 09:33:54
%S A083917 0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,1,0,0,0,0,0,1,1,0,0,0,0,0,1,
%T A083917 1,0,1,0,0,0,0,1,0,0,0,0,1,0,1,0,1,0,0,1,0,1,1,0,0,0,0,0,1,0,0,0,1,1,
%U A083917 0,1,0,0,0,1,0,0,2,0,0,0,1,0,0,1,1,0,1,0,0,0,1,0,0,1,0,0,1,1,0,0,0,1,0,0,1
%N A083917 Number of divisors of n that are congruent to 7 modulo 10.
%H A083917 Amiram Eldar, <a href="/A083917/b083917.txt">Table of n, a(n) for n = 1..10000</a>
%H A083917 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 A083917 a(n) = A000005(n) - A083910(n) - A083911(n) - A083912(n) - A083913(n) - A083914(n) - A083915(n) - A083916(n) - A083918(n) - A083919(n).
%F A083917 G.f.: Sum_{k>=1} x^(7*k)/(1 - x^(10*k)). - _Ilya Gutkovskiy_, Sep 11 2019
%F A083917 Sum_{k=1..n} a(k) = n*log(n)/10 + c*n + O(n^(1/3)*log(n)), where c = gamma(7,10) - (1 - gamma)/10 = -0.150534..., gamma(7,10) = -(psi(7/10) + 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 A083917 Table[Count[Divisors[n],_?(Mod[#,10]==7&)],{n,110}] (* _Harvey P. Dale_, Sep 15 2023 *)
%t A083917 a[n_] := DivisorSum[n, 1 &, Mod[#, 10] == 7 &]; Array[a, 100] (* _Amiram Eldar_, Dec 30 2023 *)
%o A083917 (PARI) a(n) = sumdiv(n, d, d % 10 == 7); \\ _Amiram Eldar_, Dec 30 2023
%Y A083917 Cf. A000005, A001227, A010879.
%Y A083917 Cf. A001620, A002392, A354643 (psi(7/10)).
%Y A083917 Cf. A083910, A083911, A083912, A083913, A083914, A083915, A083916, A083918, A083919.
%K A083917 nonn,easy
%O A083917 1,77
%A A083917 _Reinhard Zumkeller_, May 08 2003