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Number of twice-factorizations of n. Number of ways to choose a postpositive factorization of each part of a postpositive factorization of n.
90

%I #13 Oct 22 2017 01:38:18

%S 1,1,3,1,3,1,6,3,3,1,9,1,3,3,15,1,9,1,9,3,3,1,23,3,3,6,9,1,12,1,28,3,

%T 3,3,32,1,3,3,23,1,12,1,9,9,3,1,58,3,9,3,9,1,23,3,23,3,3,1,41,1,3,9,

%U 66,3,12,1,9,3,12,1,84,1,3,9,9,3,12,1,58,15,3

%N Number of twice-factorizations of n. Number of ways to choose a postpositive factorization of each part of a postpositive factorization of n.

%C A postpositive number is a positive integer other than 1. A postpositive factorization of n is a finite orderless sequence of postpositive numbers whose product is n.

%H Michael De Vlieger, <a href="/A281113/b281113.txt">Table of n, a(n) for n = 2..30030</a>

%H Michael De Vlieger, <a href="/A281113/a281113.txt">Indices of records in A281113</a>.

%e The a(20)=9 twice-factorizations are: ((20)), ((2*10)), ((4*5)), ((2*2*5)), ((2)*(10)), ((2)*(2*5)), ((4)*(5)), ((2*2)*(5)), ((2)*(2)*(5)).

%e Twice-factorizations of 32 organized by composite:

%e ((2)(2)(2)(2)(2)) ((2)(2)(2)(2 2)) ((2)(2)(2 2 2)) ((2)(2 2)(2 2)) ((2)(2 2 2 2)) ((2 2)(2 2 2)) ((2 2 2 2 2))

%e ((2)(2)(2)(4)) ((2)(2)(2 4)) ((2)(2 2)(4)) ((2)(4)(2 2)) ((2)(2 2 4)) ((2 2)(2 4)) ((4)(2 2 2)) ((2 2 2 4))

%e ((2)(2)(8)) ((2)(2 8)) ((2 2)(8)) ((2 2 8))

%e ((2)(4)(4)) ((2)(4 4)) ((4)(2 4)) ((2 4 4))

%e ((2)(16)) ((2 16))

%e ((4)(8)) ((4 8))

%e ((32)).

%e Twice-factorizations of 32 organized by domain:

%e ((2)(2)(2)(2)(2))

%e ((2)(2)(2)(2 2)) ((2)(2)(2)(4))

%e ((2)(2)(2 2 2)) ((2)(2)(2 4)) ((2)(2)(8))

%e ((2)(2 2)(2 2)) ((2)(2 2)(4)) ((2)(4)(2 2)) ((2)(4)(4))

%e ((2)(2 2 2 2)) ((2)(2 2 4)) ((2)(2 8)) ((2)(4 4)) ((2)(16))

%e ((2 2)(2 2 2)) ((2 2)(2 4)) ((2 2)(8)) ((4)(2 2 2)) ((4)(2 4)) ((4)(8))

%e ((2 2 2 2 2)) ((2 2 2 4)) ((2 2 8)) ((2 4 4)) ((2 16)) ((4 8)) ((32)).

%t postfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[postfacs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];

%t twicefacs[n_]:=Join@@Tuples/@Map[postfacs,postfacs[n],{2}];

%t Table[Length[twicefacs[n]],{n,2,24}]

%Y Cf. A001055(n) = number of factorizations of n, A050336(n) = number of orderless twice-factorizations of n, A162247(n) = factors of factorizations of n, A063834(n) = a(p^(n-1)), A007716, A269134, A281116.

%K nonn

%O 2,3

%A _Gus Wiseman_, Jan 14 2017