A339318 Dirichlet g.f.: Product_{k>=2} 1 / (1 - k^(-s))^3.

1, 3, 3, 9, 3, 12, 3, 22, 9, 12, 3, 39, 3, 12, 12, 51, 3, 39, 3, 39, 12, 12, 3, 105, 9, 12, 22, 39, 3, 57, 3, 108, 12, 12, 12, 135, 3, 12, 12, 105, 3, 57, 3, 39, 39, 12, 3, 258, 9, 39, 12, 39, 3, 105, 12, 105, 12, 12, 3, 201, 3, 12, 39, 221, 12, 57, 3, 39, 12, 57

Offset: 1

Author

Ilya Gutkovskiy, Nov 30 2020

Keywords

• nonn

Comments

Number of factorizations of n into factors (greater than 1) of 3 kinds.

Examples

			From _Antti Karttunen_, Dec 15 2021: (Start)
For n = 8, A001055(8) = 3, as it has three ordinary factorizations: (8), (4*2), (2*2*2). When allowing each of the factors appear in three different guises (here indicated with a subscript), and where neither the order of factors nor their subscripts matter, we get the following 22 different factorizations:
  (8_3), (8_2), (8_1),
  (4_3 * 2_3), (4_3 * 2_2), (4_3 * 2_1),
  (4_2 * 2_3), (4_2 * 2_2), (4_2 * 2_1),
  (4_1 * 2_3), (4_1 * 2_2), (4_1 * 2_1),
  (2_3 * 2_3 * 2_3), (2_3 * 2_3 * 2_2), (2_3 * 2_3 * 2_1),
  (2_3 * 2_2 * 2_2), (2_3 * 2_2 * 2_1), (2_3 * 2_1 * 2_1),
  (2_2 * 2_2 * 2_2), (2_2 * 2_2 * 2_1), (2_2 * 2_1 * 2_1),
  (2_1 * 2_1 * 2_1),
therefore a(8) = 22. (End)
		

Links

• David A. Corneth, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000716, A001055, A301830, A339319, A339320, A339321, A339322, A339323, A339324.

Programs

PARI

A339318list(n) = MultEulerT(vector(n, i, 3)); \\ Antti Karttunen, Jan 21 2022, using Andrew Howroyd's program given in A301830.

Formula

a(p^k) = A000716(k) for prime p.