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A320001 Number of proper divisors of n of the form 6*k + 1.

%I A320001 #19 Nov 25 2023 07:58:46
%S A320001 0,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,2,1,1,1,1,2,1,2,1,1,1,1,1,1,
%T A320001 2,1,1,2,2,1,1,2,1,1,1,1,1,1,2,2,1,2,1,1,1,2,2,1,1,1,1,2,2,1,2,1,1,1,
%U A320001 1,2,1,1,1,2,2,2,2,2,1,1,1,1,1,2,1,2,1,1,1,1,3,1,2,1,2,1,1,3,1,2,1,1,1,2,2
%N A320001 Number of proper divisors of n of the form 6*k + 1.
%H A320001 Antti Karttunen, <a href="/A320001/b320001.txt">Table of n, a(n) for n = 1..65537</a>
%H A320001 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 A320001 a(n) = A279060(n) - [+1 = n (mod 6)], where [ ] is the Iverson bracket, giving 1 only when n = 1 mod 6, and 0 otherwise.
%F A320001 a(n) = A320015(n) - A320005(n).
%F A320001 a(n) = A007814(A319991(n)).
%F A320001 G.f.: Sum_{k>=1} x^(12*k-10) / (1 - x^(6*k-5)). - _Ilya Gutkovskiy_, Apr 14 2021
%F A320001 Sum_{k=1..n} a(k) = n*log(n)/6 + c*n + O(n^(1/3)*log(n)), where c = gamma(1,6) - (2 - gamma)/6 = 0.519597..., gamma(1,6) = -(psi(1/6) + 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 A320001 a[n_] := DivisorSum[n, 1 &, # < n && Mod[#, 6] == 1 &]; Array[a, 100] (* _Amiram Eldar_, Nov 25 2023 *)
%o A320001 (PARI) A320001(n) = if(!n,n,sumdiv(n, d, (d<n)*(1==(d%6))));
%Y A320001 Cf. A279060, A293895, A319991, A320003, A320005, A320015.
%Y A320001 Cf. A001620, A016629, A222457 (psi(1/6)).
%K A320001 nonn,easy
%O A320001 1,14
%A A320001 _Antti Karttunen_, Oct 03 2018