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A083912 Number of divisors of n that are congruent to 2 modulo 10.

%I A083912 #20 Dec 30 2023 04:57:04
%S A083912 0,1,0,1,0,1,0,1,0,1,0,2,0,1,0,1,0,1,0,1,0,2,0,2,0,1,0,1,0,1,0,2,0,1,
%T A083912 0,2,0,1,0,1,0,2,0,2,0,1,0,2,0,1,0,2,0,1,0,1,0,1,0,2,0,2,0,2,0,2,0,1,
%U A083912 0,1,0,3,0,1,0,1,0,1,0,1,0,2,0,3,0,1,0,2,0,1,0,2,0,1,0,3,0,1,0,1,0,2,0,2,0
%N A083912 Number of divisors of n that are congruent to 2 modulo 10.
%H A083912 Antti Karttunen, <a href="/A083912/b083912.txt">Table of n, a(n) for n = 1..65537</a>
%H A083912 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 A083912 a(n) = A000005(n) - A083910(n) - A083911(n) - A083913(n) - A083914(n) - A083915(n) - A083916(n) - A083917(n) - A083918(n) - A083919(n).
%F A083912 G.f.: Sum_{k>=1} x^(2*k)/(1 - x^(10*k)). - _Ilya Gutkovskiy_, Sep 11 2019
%F A083912 Sum_{k=1..n} a(k) = n*log(n)/10 + c*n + O(n^(1/3)*log(n)), where c = gamma(2,10) - (1 - gamma)/10 = 0.256367..., gamma(2,10) = -(psi(1/5) + 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 A083912 a[n_] := Sum[If[Mod[d, 10] == 2, 1, 0], {d, Divisors[n]}];
%t A083912 Array[a, 105] (* _Jean-François Alcover_, Dec 02 2021 *)
%t A083912 a[n_] := DivisorSum[n, 1 &, Mod[#, 10] == 2 &]; Array[a, 100] (* _Amiram Eldar_, Dec 30 2023 *)
%o A083912 (PARI) A083912(n) = sumdiv(n,d,2==(d%10)); \\ _Antti Karttunen_, Jan 22 2020
%Y A083912 Cf. A000005, A001227, A010879.
%Y A083912 Cf. A001620, A002392, A200135 (psi(1/5)).
%Y A083912 Cf. A083910, A083911, A083913, A083914, A083915, A083916, A083917, A083918, A083919.
%K A083912 nonn,easy
%O A083912 1,12
%A A083912 _Reinhard Zumkeller_, May 08 2003