A296121 Number of twice-factorizations of n with no repeated factorizations.

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

Offset: 1

Author

Gus Wiseman, Dec 05 2017

Keywords

• nonn

Comments

From Robert G. Wilson v, Dec 05 2017: (Start)

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

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

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

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

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

(End)

Examples

			The a(12) = 8 twice-factorizations:
(2)*(2*3), (3)*(2*2), (2*2*3),
(2)*(6), (2*6),
(3)*(4), (3*4),
(12).
		

Links

• Robert G. Wilson v, Table of n, a(n) for n = 1..1000

Crossrefs

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

Programs

Mathematica

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Join@@Table[Select[Tuples[facs/@p],UnsameQ@@#&],{p,facs[n]}]],{n,100}]