%I #19 Jan 27 2022 21:15:22

%S 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,

%T 39,3,57,3,108,12,12,12,135,3,12,12,105,3,57,3,39,39,12,3,258,9,39,12,

%U 39,3,105,12,105,12,12,3,201,3,12,39,221,12,57,3,39,12,57

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

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

%H David A. Corneth, <a href="/A339318/b339318.txt">Table of n, a(n) for n = 1..10000</a>

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

%e From _Antti Karttunen_, Dec 15 2021: (Start)

%e 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:

%e (8_3), (8_2), (8_1),

%e (4_3 * 2_3), (4_3 * 2_2), (4_3 * 2_1),

%e (4_2 * 2_3), (4_2 * 2_2), (4_2 * 2_1),

%e (4_1 * 2_3), (4_1 * 2_2), (4_1 * 2_1),

%e (2_3 * 2_3 * 2_3), (2_3 * 2_3 * 2_2), (2_3 * 2_3 * 2_1),

%e (2_3 * 2_2 * 2_2), (2_3 * 2_2 * 2_1), (2_3 * 2_1 * 2_1),

%e (2_2 * 2_2 * 2_2), (2_2 * 2_2 * 2_1), (2_2 * 2_1 * 2_1),

%e (2_1 * 2_1 * 2_1),

%e therefore a(8) = 22. (End)

%o (PARI) A339318list(n) = MultEulerT(vector(n, i, 3)); \\ _Antti Karttunen_, Jan 21 2022, using _Andrew Howroyd_'s program given in A301830.

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

%K nonn

%O 1,2

%A _Ilya Gutkovskiy_, Nov 30 2020