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A083911 Number of divisors of n that are congruent to 1 modulo 10.

%I A083911 #13 Dec 30 2023 04:56:13
%S A083911 1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,2,2,1,1,1,1,1,1,1,1,2,1,2,1,
%T A083911 1,1,1,1,1,1,2,2,1,2,1,1,1,1,1,1,2,1,1,1,2,1,1,1,1,1,2,2,2,1,1,2,1,1,
%U A083911 1,1,2,1,1,1,1,1,2,1,1,1,2,2,1,2,1,1,1,2,1,1,2,1,2,1,1,1,1,1,2,1,2,2,1,1,2
%N A083911 Number of divisors of n that are congruent to 1 modulo 10.
%H A083911 Amiram Eldar, <a href="/A083911/b083911.txt">Table of n, a(n) for n = 1..10000</a>
%H A083911 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 A083911 a(n) = A000005(n) - A083910(n) - A083912(n) - A083913(n) - A083914(n) - A083915(n) - A083916(n) - A083917(n) - A083918(n) - A083919(n).
%F A083911 G.f.: Sum_{k>=1} x^k/(1 - x^(10*k)). - _Ilya Gutkovskiy_, Sep 11 2019
%F A083911 Sum_{k=1..n} a(k) = n*log(n)/10 + c*n + O(n^(1/3)*log(n)), where c = gamma(1,10) - (1 - gamma)/10 = 0.769838..., gamma(1,6) = -(psi(1/10) + 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 A083911 a[n_] := DivisorSum[n, 1 &, Mod[#, 10] == 1 &]; Array[a, 100] (* _Amiram Eldar_, Dec 30 2023 *)
%o A083911 (PARI) a(n) = sumdiv(n, d, d % 10 == 1); \\ _Amiram Eldar_, Dec 30 2023
%Y A083911 Cf. A000005, A001227, A010879.
%Y A083911 Cf. A001620, A002392, A306716 (psi(1/10)).
%Y A083911 Cf. A083910, A083912, A083913, A083914, A083915, A083916, A083917, A083918, A083919.
%K A083911 nonn,easy
%O A083911 1,11
%A A083911 _Reinhard Zumkeller_, May 08 2003