cp's OEIS Frontend

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

%I A281113 #13 Oct 22 2017 01:38:18
%S A281113 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 A281113 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 A281113 66,3,12,1,9,3,12,1,84,1,3,9,9,3,12,1,58,15,3
%N A281113 Number of twice-factorizations of n. Number of ways to choose a postpositive factorization of each part of a postpositive factorization of n.
%C A281113 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 A281113 Michael De Vlieger, <a href="/A281113/b281113.txt">Table of n, a(n) for n = 2..30030</a>
%H A281113 Michael De Vlieger, <a href="/A281113/a281113.txt">Indices of records in A281113</a>.
%e A281113 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 A281113 Twice-factorizations of 32 organized by composite:
%e A281113 ((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 A281113 ((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 A281113 ((2)(2)(8))       ((2)(2 8))       ((2 2)(8))      ((2 2 8))
%e A281113 ((2)(4)(4))       ((2)(4 4))       ((4)(2 4))      ((2 4 4))
%e A281113 ((2)(16))         ((2 16))
%e A281113 ((4)(8))          ((4 8))
%e A281113 ((32)).
%e A281113 Twice-factorizations of 32 organized by domain:
%e A281113 ((2)(2)(2)(2)(2))
%e A281113 ((2)(2)(2)(2 2)) ((2)(2)(2)(4))
%e A281113 ((2)(2)(2 2 2))  ((2)(2)(2 4)) ((2)(2)(8))
%e A281113 ((2)(2 2)(2 2))  ((2)(2 2)(4)) ((2)(4)(2 2)) ((2)(4)(4))
%e A281113 ((2)(2 2 2 2))   ((2)(2 2 4))  ((2)(2 8))    ((2)(4 4))   ((2)(16))
%e A281113 ((2 2)(2 2 2))   ((2 2)(2 4))  ((2 2)(8))    ((4)(2 2 2)) ((4)(2 4)) ((4)(8))
%e A281113 ((2 2 2 2 2))    ((2 2 2 4))   ((2 2 8))     ((2 4 4))    ((2 16))   ((4 8)) ((32)).
%t A281113 postfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[postfacs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A281113 twicefacs[n_]:=Join@@Tuples/@Map[postfacs,postfacs[n],{2}];
%t A281113 Table[Length[twicefacs[n]],{n,2,24}]
%Y A281113 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 A281113 nonn
%O A281113 2,3
%A A281113 _Gus Wiseman_, Jan 14 2017