%I #14 Dec 30 2023 09:33:54

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

%H Amiram Eldar, <a href="/A083917/b083917.txt">Table of n, a(n) for n = 1..10000</a>

%H 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 a(n) = A000005(n) - A083910(n) - A083911(n) - A083912(n) - A083913(n) - A083914(n) - A083915(n) - A083916(n) - A083918(n) - A083919(n).

%F G.f.: Sum_{k>=1} x^(7*k)/(1 - x^(10*k)). - _Ilya Gutkovskiy_, Sep 11 2019

%F 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 Table[Count[Divisors[n],_?(Mod[#,10]==7&)],{n,110}] (* _Harvey P. Dale_, Sep 15 2023 *)

%t a[n_] := DivisorSum[n, 1 &, Mod[#, 10] == 7 &]; Array[a, 100] (* _Amiram Eldar_, Dec 30 2023 *)

%o (PARI) a(n) = sumdiv(n, d, d % 10 == 7); \\ _Amiram Eldar_, Dec 30 2023

%Y Cf. A000005, A001227, A010879.

%Y Cf. A001620, A002392, A354643 (psi(7/10)).

%Y Cf. A083910, A083911, A083912, A083913, A083914, A083915, A083916, A083918, A083919.

%K nonn,easy

%O 1,77

%A _Reinhard Zumkeller_, May 08 2003