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Number of twice-factorizations of n with no repeated factorizations.
5

%I #21 Sep 20 2018 00:31:25

%S 1,1,1,2,1,3,1,5,2,3,1,8,1,3,3,10,1,8,1,8,3,3,1,20,2,3,5,8,1,12,1,20,

%T 3,3,3,25,1,3,3,20,1,12,1,8,8,3,1,47,2,8,3,8,1,20,3,20,3,3,1,38,1,3,8,

%U 40,3,12,1,8,3,12,1,68,1,3,8,8,3,12,1,47,10

%N Number of twice-factorizations of n with no repeated factorizations.

%C From _Robert G. Wilson v_, Dec 05 2017: (Start)

%C a(n) = 1 iff n equals 1 or is a prime;

%C a(n) = 2 iff n is a prime squared;

%C a(n) = 3 iff n is a squarefree semiprime;

%C a(n) = 5 iff n is a prime cube;

%C a(n) = 8 iff n is of the form p^2*q, etc.

%C (End)

%H Robert G. Wilson v, <a href="/A296121/b296121.txt">Table of n, a(n) for n = 1..1000</a>

%e The a(12) = 8 twice-factorizations:

%e (2)*(2*3), (3)*(2*2), (2*2*3),

%e (2)*(6), (2*6),

%e (3)*(4), (3*4),

%e (12).

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

%t Table[Length[Join@@Table[Select[Tuples[facs/@p],UnsameQ@@#&],{p,facs[n]}]],{n,100}]

%Y Cf. A001055, A045778, A050345, A063834, A089723, A281113, A296118, A296119, A296120, A296122.

%K nonn

%O 1,4

%A _Gus Wiseman_, Dec 05 2017