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A083914 Number of divisors of n that are congruent to 4 modulo 10.

%I A083914 #20 Dec 30 2023 09:33:59
%S A083914 0,0,0,1,0,0,0,1,0,0,0,1,0,1,0,1,0,0,0,1,0,0,0,2,0,0,0,2,0,0,0,1,0,1,
%T A083914 0,1,0,0,0,1,0,1,0,2,0,0,0,2,0,0,0,1,0,1,0,2,0,0,0,1,0,0,0,2,0,0,0,2,
%U A083914 0,1,0,2,0,1,0,1,0,0,0,1,0,0,0,3,0,0,0,2,0,0,0,1,0,1,0,2,0,1,0,1,0,1,0,2,0
%N A083914 Number of divisors of n that are congruent to 4 modulo 10.
%H A083914 Antti Karttunen, <a href="/A083914/b083914.txt">Table of n, a(n) for n = 1..10000</a>
%H A083914 Antti Karttunen, <a href="/A083914/a083914.txt">Data supplement: n, a(n) computed for n =  1..100000</a>
%H A083914 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 A083914 a(n) = A000005(n) - A083910(n) - A083911(n) - A083912(n) - A083913(n) - A083915(n) - A083916(n) - A083917(n) - A083918(n) - A083919(n).
%F A083914 G.f.: Sum_{k>=1} x^(4*k)/(1 - x^(10*k)). - _Ilya Gutkovskiy_, Sep 11 2019
%F A083914 Sum_{k=1..n} a(k) = n*log(n)/10 + c*n + O(n^(1/3)*log(n)), where c = gamma(4,10) - (1 - gamma)/10 = -0.0163984..., gamma(4,10) = -(psi(2/5) + 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 A083914 Table[Count[Divisors[n],_?(Mod[#,10]==4&)],{n,110}] (* _Harvey P. Dale_, Dec 09 2014 *)
%t A083914 a[n_] := DivisorSum[n, 1 &, Mod[#, 10] == 4 &]; Array[a, 100] (* _Amiram Eldar_, Dec 30 2023 *)
%o A083914 (PARI) A083914(n) = sumdiv(n,d,(4==(d%10))); \\ _Antti Karttunen_, Nov 07 2018
%Y A083914 Cf. A000005, A001227, A010879.
%Y A083914 Cf. A001620, A002392, A200136 (psi(2/5)).
%Y A083914 Cf. A083910, A083911, A083912, A083913, A083915, A083916, A083917, A083918, A083919.
%K A083914 nonn,easy
%O A083914 1,24
%A A083914 _Reinhard Zumkeller_, May 08 2003