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 A301598 Number of thrice-factorizations of n. 3
 1, 1, 1, 4, 1, 4, 1, 10, 4, 4, 1, 16, 1, 4, 4, 34, 1, 16, 1, 16, 4, 4, 1, 54, 4, 4, 10, 16, 1, 22, 1, 80, 4, 4, 4, 78, 1, 4, 4, 54, 1, 22, 1, 16, 16, 4, 1, 181, 4, 16, 4, 16, 1, 54, 4, 54, 4, 4, 1, 102, 1, 4, 16, 254, 4, 22, 1, 16, 4, 22, 1, 272, 1, 4, 16, 16 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS A thrice-factorization of n is a choice of a twice-factorization of each factor in a factorization of n. Thrice-factorizations correspond to intervals in the lattice form of the multiorder of integer factorizations. LINKS Gus Wiseman, The (2*2*3*3) component, version 2. FORMULA Dirichlet g.f.: Product_{n > 1} 1/(1 - A281113(n)/n^s). EXAMPLE The a(12) = 16 thrice-factorizations: ((2))*((2))*((3)), ((2))*((2)*(3)), ((3))*((2)*(2)), ((2)*(2)*(3)), ((2))*((2*3)), ((2)*(2*3)), ((2))*((6)), ((2)*(6)), ((3))*((2*2)), ((3)*(2*2)), ((3))*((4)), ((3)*(4)), ((2*2*3)), ((2*6)), ((3*4)), ((12)). MATHEMATICA facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]]; twifacs[n_]:=Join@@Table[Tuples[facs/@f], {f, facs[n]}]; thrifacs[n_]:=Join@@Table[Tuples[twifacs/@f], {f, facs[n]}]; Table[Length[thrifacs[n]], {n, 15}] CROSSREFS Cf. A001055, A007716, A050336, A050338, A063834, A162247, A269134, A281113, A281116, A301595, A301598, A301706. Sequence in context: A097936 A277027 A050338 * A077088 A297420 A156896 Adjacent sequences:  A301595 A301596 A301597 * A301599 A301600 A301601 KEYWORD nonn AUTHOR Gus Wiseman, Mar 24 2018 STATUS approved

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Last modified June 25 21:43 EDT 2019. Contains 324357 sequences. (Running on oeis4.)