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A319995 Number of divisors of n of the form 6*k + 5.

%I A319995 #17 Nov 25 2023 04:32:10
%S A319995 0,0,0,0,1,0,0,0,0,1,1,0,0,0,1,0,1,0,0,1,0,1,1,0,1,0,0,0,1,1,0,0,1,1,
%T A319995 2,0,0,0,0,1,1,0,0,1,1,1,1,0,0,1,1,0,1,0,2,0,0,1,1,1,0,0,0,0,2,1,0,1,
%U A319995 1,2,1,0,0,0,1,0,2,0,0,1,0,1,1,0,2,0,1,1,1,1,0,1,0,1,2,0,0,0,1,1,1,1,0,0,2
%N A319995 Number of divisors of n of the form 6*k + 5.
%H A319995 Antti Karttunen, <a href="/A319995/b319995.txt">Table of n, a(n) for n = 1..65537</a>
%H A319995 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 A319995 a(n) = A035218(n) - A279060(n).
%F A319995 G.f.: Sum_{k>=1} x^(5*k)/(1 - x^(6*k)). - _Ilya Gutkovskiy_, Sep 11 2019
%F A319995 Sum_{k=1..n} a(k) = n*log(n)/6 + c*n + O(n^(1/3)*log(n)), where c = gamma(5,6) - (1 - gamma)/6 = -0.220635..., 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 A319995 a[n_] := DivisorSum[n, 1 &, Mod[#, 6] == 5 &]; Array[a, 100] (* _Amiram Eldar_, Nov 25 2023 *)
%o A319995 (PARI) A319995(n) = if(!n,n,sumdiv(n, d, (5==(d%6))));
%Y A319995 Cf. A035218, A279060, A320005.
%Y A319995 Cf. A001620, A016629, A222458 (psi(5/6)).
%K A319995 nonn,easy
%O A319995 1,35
%A A319995 _Antti Karttunen_, Oct 03 2018