A281113 Number of twice-factorizations of n. Number of ways to choose a postpositive factorization of each part of a postpositive factorization of n.

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

Offset: 2

Author

Gus Wiseman, Jan 14 2017

Keywords

• nonn

Comments

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.

Examples

			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)).
Twice-factorizations of 32 organized by composite:
((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))
((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))
((2)(2)(8))       ((2)(2 8))       ((2 2)(8))      ((2 2 8))
((2)(4)(4))       ((2)(4 4))       ((4)(2 4))      ((2 4 4))
((2)(16))         ((2 16))
((4)(8))          ((4 8))
((32)).
Twice-factorizations of 32 organized by domain:
((2)(2)(2)(2)(2))
((2)(2)(2)(2 2)) ((2)(2)(2)(4))
((2)(2)(2 2 2))  ((2)(2)(2 4)) ((2)(2)(8))
((2)(2 2)(2 2))  ((2)(2 2)(4)) ((2)(4)(2 2)) ((2)(4)(4))
((2)(2 2 2 2))   ((2)(2 2 4))  ((2)(2 8))    ((2)(4 4))   ((2)(16))
((2 2)(2 2 2))   ((2 2)(2 4))  ((2 2)(8))    ((4)(2 2 2)) ((4)(2 4)) ((4)(8))
((2 2 2 2 2))    ((2 2 2 4))   ((2 2 8))     ((2 4 4))    ((2 16))   ((4 8)) ((32)).
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 2..30030
• Michael De Vlieger, Indices of records in A281113.

Crossrefs

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.

Programs

Mathematica

postfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[postfacs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
twicefacs[n_]:=Join@@Tuples/@Map[postfacs,postfacs[n],{2}];
Table[Length[twicefacs[n]],{n,2,24}]