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

1,4

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)

Links

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

Example

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

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}]

Crossrefs

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

Sequence in context: A278136 A085053 A296118 * A277120 A104725 A289079

Adjacent sequences: A296118 A296119 A296120 * A296122 A296123 A296124

Keyword

nonn

Author

Gus Wiseman, Dec 05 2017

Status

approved