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Number of proper divisors of n of the form 3k+2.
7

%I #17 Nov 25 2023 08:03:59

%S 0,0,0,1,0,1,0,1,0,2,0,1,0,1,1,2,0,1,0,2,0,2,0,2,1,1,0,2,0,2,0,2,1,2,

%T 1,1,0,1,0,4,0,2,0,2,1,2,0,2,0,2,1,2,0,1,2,3,0,2,0,3,0,1,0,3,1,2,0,2,

%U 1,4,0,2,0,1,1,2,1,2,0,4,0,2,0,2,2,1,1,4,0,2,0,2,0,2,1,3,0,2,1,4,0,2,0,3,2

%N Number of proper divisors of n of the form 3k+2.

%H Antti Karttunen, <a href="/A293896/b293896.txt">Table of n, a(n) for n = 1..20000</a>

%H 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 a(n) = A001822(n) - [n == 2 (mod 3)].

%F G.f.: Sum_{k>=1} x^(6*k-2) / (1 - x^(3*k-1)). - _Ilya Gutkovskiy_, Apr 14 2021

%F Sum_{k=1..n} a(k) = n*log(n)/3 + c*n + O(n^(1/3)*log(n)), where c = gamma(2,3) - (2 - gamma)/3 = A256843 - (2 - A001620)/3 = -0.401054... (Smith and Subbarao, 1981). - _Amiram Eldar_, Nov 25 2023

%t Table[DivisorSum[n, 1 &, And[Mod[#, 3] == 2, # != n] &], {n, 105}] (* _Michael De Vlieger_, Nov 08 2017 *)

%o (PARI) A293896(n) = sumdiv(n,d,(d<n)*(2==(d%3)));

%Y Cf. A001620, A001822, A256843, A293895, A293898, A293899.

%K nonn,easy

%O 1,10

%A _Antti Karttunen_, Nov 06 2017