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).

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, 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, 0, 0, 0, 0, 0, 9291, 1668, 0, 2170, 226, 0, 1052, 59, 0

Offset: 10

Author

Markus Sigg, Jul 15 2022

Keywords

• nonn

Comments

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).

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.

a(n) > 0 iff n is in A355631.

Links

• Markus Sigg, Table of n, a(n) for n = 10..243

Crossrefs

Cf. A355626, A355628, A355629, A355630, A355631.

Programs

PARI

\\ See A355626.