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A320005 Number of proper divisors of n of the form 6*k + 5.

%I A320005 #17 Nov 25 2023 07:59:16
%S A320005 0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,1,0,0,1,0,0,0,0,1,0,0,1,1,
%T A320005 1,0,0,0,0,1,0,0,0,1,1,1,0,0,0,1,1,0,0,0,2,0,0,1,0,1,0,0,0,0,1,1,0,1,
%U A320005 1,2,0,0,0,0,1,0,1,0,0,1,0,1,0,0,2,0,1,1,0,1,0,1,0,1,1,0,0,0,1,1,0,1,0,0,2
%N A320005 Number of proper divisors of n of the form 6*k + 5.
%H A320005 Antti Karttunen, <a href="/A320005/b320005.txt">Table of n, a(n) for n = 1..65537</a>
%H A320005 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 A320005 a(n) = A319995(n) - [+5 = n (mod 6)], where [ ] is the Iverson bracket, giving 1 only when n = -1 mod 6, and 0 otherwise.
%F A320005 a(n) = A320015(n) - A320001(n).
%F A320005 a(n) = A007814(A319992(n)).
%F A320005 G.f.: Sum_{k>=1} x^(12*k-2) / (1 - x^(6*k-1)). - _Ilya Gutkovskiy_, Apr 14 2021
%F A320005 Sum_{k=1..n} a(k) = n*log(n)/6 + c*n + O(n^(1/3)*log(n)), where c = gamma(5,6) - (2 - gamma)/6 = -0.387302..., gamma(5,6) = -(psi(5/6) + log(6))/6 is a generalized Euler constant, and gamma is Euler's constant (A001620) (Smith and Subbarao, 1981). - _Amiram Eldar_, Nov 25 2023
%t A320005 a[n_] := DivisorSum[n, 1 &, # < n && Mod[#, 6] == 5 &]; Array[a, 100] (* _Amiram Eldar_, Nov 25 2023 *)
%o A320005 (PARI) A320005(n) = if(!n,n,sumdiv(n, d, (d<n)*(5==(d%6))));
%Y A320005 Cf. A293896, A319992, A319995, A320001, A320003, A320015.
%Y A320005 Cf. A001620, A016629, A222458 (psi(5/6)).
%K A320005 nonn,easy
%O A320005 1,55
%A A320005 _Antti Karttunen_, Oct 03 2018