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A293895 Number of proper divisors of n of the form 3k+1.

%I A293895 #18 Nov 25 2023 08:03:37
%S A293895 0,1,1,1,1,1,1,2,1,1,1,2,1,2,1,2,1,1,1,3,2,1,1,2,1,2,1,3,1,2,1,3,1,1,
%T A293895 2,2,1,2,2,3,1,2,1,3,1,1,1,3,2,3,1,3,1,1,1,4,2,1,1,3,1,2,2,3,2,2,1,3,
%U A293895 1,3,1,2,1,2,2,3,2,2,1,5,1,1,1,4,1,2,1,3,1,2,3,3,2,1,2,3,1,3,1,4,1,2,1,4,2
%N A293895 Number of proper divisors of n of the form 3k+1.
%H A293895 Antti Karttunen, <a href="/A293895/b293895.txt">Table of n, a(n) for n = 1..20000</a>
%H A293895 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 A293895 a(n) = A001817(n) - [n == 1 (mod 3)].
%F A293895 G.f.: Sum_{k>=1} x^(6*k-4) / (1 - x^(3*k-2)). - _Ilya Gutkovskiy_, Apr 14 2021
%F A293895 Sum_{k=1..n} a(k) = n*log(n)/3 + c*n + O(n^(1/3)*log(n)), where c = gamma(1,3) - (2 - gamma)/3 = A256425 - (2 - A001620)/3 = 0.203545... (Smith and Subbarao, 1981). - _Amiram Eldar_, Nov 25 2023
%t A293895 Table[DivisorSum[n, 1 &, And[Mod[#, 3] == 1, # != n] &], {n, 105}] (* _Michael De Vlieger_, Nov 08 2017 *)
%o A293895 (PARI) A293895(n) = sumdiv(n,d,(d<n)*(1==(d%3)));
%Y A293895 Cf. A001620, A001817, A256425, A293451, A293896, A293897, A293899.
%K A293895 nonn,easy
%O A293895 1,8
%A A293895 _Antti Karttunen_, Nov 06 2017