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A355627 a(n) is the number of tuples (t_1, ..., t_k) with a positive integer k and integers 2 <= t_1 <= ... <= t_k such that n = Product_{i = 1..k} (3 + 1/t_i).

%I A355627 #19 Mar 22 2025 23:34:31
%S A355627 2,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,50,14,0,2,9,0,2,2,0,1,0,0,0,0,0,
%T A355627 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%U A355627 0,0,0,0,0,9291,1668,0,2170,226,0,1052,59,0
%N A355627 a(n) is the number of tuples (t_1, ..., t_k) with a positive integer k and integers 2 <= t_1 <= ... <= t_k such that n = Product_{i = 1..k} (3 + 1/t_i).
%C A355627 Because 3^k < Product_{i = 1..k} (3 + 1/t_i) < 3.5^k, a(n) > 0 is possible only for 10 <= n <= 12 (k = 2), 28 <= n <= 42 (k = 3), 82 <= n <= 150 (k = 4), 244 <= n <= 525 (k = 5) etc. For n <= 19683, there can exist at most one k such that n can be written as a product of k factors (3 + 1/t_i).
%C A355627 a(n) = 0 when n is a multiple of 3: Suppose n = Product_{i = 1..k} (3 + 1/t_i). Then n * Product_{i = 1..k} t_i = Product_{i = 1..k} (3 * t_i + 1). The right hand side is not a multiple of 3, so neither n nor any of the t_i can be a multiple of 3.
%C A355627 a(n) > 0 iff n is in A355631.
%H A355627 Markus Sigg, <a href="/A355627/b355627.txt">Table of n, a(n) for n = 10..243</a>
%o A355627 (PARI) \\ See A355626.
%Y A355627 Cf. A355626, A355628, A355629, A355630, A355631.
%K A355627 nonn
%O A355627 10,1
%A A355627 _Markus Sigg_, Jul 15 2022