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A339318
Dirichlet g.f.: Product_{k>=2} 1 / (1 - k^(-s))^3.
10
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
1,2
COMMENTS
Number of factorizations of n into factors (greater than 1) of 3 kinds.
LINKS
FORMULA
a(p^k) = A000716(k) for prime p.
EXAMPLE
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)
PROG
(PARI) A339318list(n) = MultEulerT(vector(n, i, 3)); \\ Antti Karttunen, Jan 21 2022, using Andrew Howroyd's program given in A301830.
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Nov 30 2020
STATUS
approved