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A083919 Number of divisors of n that are congruent to 9 modulo 10.

%I A083919 #12 Dec 30 2023 09:34:03
%S A083919 0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,
%T A083919 0,1,0,1,1,0,0,0,0,0,1,0,0,0,1,0,0,0,0,1,0,0,1,1,1,0,0,0,1,0,0,0,0,0,
%U A083919 1,0,0,1,0,0,0,1,0,1,1,0,1,0,0,0,0,0,1,0,1,1,0,0,0,0,1,0,0,1,2,0,0,0,0,0,0
%N A083919 Number of divisors of n that are congruent to 9 modulo 10.
%H A083919 Amiram Eldar, <a href="/A083919/b083919.txt">Table of n, a(n) for n = 1..10000</a>
%H A083919 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 A083919 a(n) = A000005(n) - A083910(n) - A083911(n) - A083912(n) - A083913(n) - A083914(n) - A083915(n) - A083916(n) - A083917(n) - A083918(n).
%F A083919 G.f.: Sum_{k>=1} x^(9*k)/(1 - x^(10*k)). - _Ilya Gutkovskiy_, Sep 11 2019
%F A083919 Sum_{k=1..n} a(k) = n*log(n)/10 + c*n + O(n^(1/3)*log(n)), where c = gamma(9,10) - (1 - gamma)/10 = -0.197044..., gamma(9,10) = -(psi(9/10) + 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 A083919 a[n_] := DivisorSum[n, 1 &, Mod[#, 10] == 9 &]; Array[a, 100] (* _Amiram Eldar_, Dec 30 2023 *)
%o A083919 (PARI) a(n) = sumdiv(n, d, d % 10 == 9); \\ _Amiram Eldar_, Dec 30 2023
%Y A083919 Cf. A000005, A001227, A010879.
%Y A083919 Cf. A001620, A002392, A354644 (psi(9/10)).
%Y A083919 Cf. A083910, A083911, A083912, A083913, A083914, A083915, A083916, A083917, A083918.
%K A083919 nonn,easy
%O A083919 1,99
%A A083919 _Reinhard Zumkeller_, May 08 2003