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

2,3

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.

Links

Michael De Vlieger, Table of n, a(n) for n = 2..30030

Michael De Vlieger, Indices of records in A281113.

Example

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)).

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}]

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.

Sequence in context: A066637 A317144 A050336 * A361519 A236757 A095250

Adjacent sequences: A281110 A281111 A281112 * A281114 A281115 A281116

Keyword

nonn

Author

Gus Wiseman, Jan 14 2017

Status

approved