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A296121 Number of twice-factorizations of n with no repeated factorizations. 5
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified July 28 00:54 EDT 2021. Contains 346316 sequences. (Running on oeis4.)