A321514 - OEIS

A321514

Number of ways to choose a factorization of each integer from 2 to n into factors > 1.

1, 1, 1, 2, 2, 4, 4, 12, 24, 48, 48, 192, 192, 384, 768, 3840, 3840, 15360, 15360, 61440, 122880, 245760, 245760, 1720320, 3440640, 6881280, 20643840, 82575360, 82575360, 412876800, 412876800, 2890137600, 5780275200, 11560550400, 23121100800, 208089907200

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OFFSET

1,4

LINKS

Table of n, a(n) for n=1..36.

FORMULA

a(n) = Product_{k = 1..n} A001055(k).

EXAMPLE

The a(8) = 12 ways to choose a factorization of each integer from 2 to 8:

(2)*(3)*(4)*(5)*(6)*(7)*(8)

(2)*(3)*(4)*(5)*(6)*(7)*(2*4)

(2)*(3)*(4)*(5)*(2*3)*(7)*(8)

(2)*(3)*(2*2)*(5)*(6)*(7)*(8)

(2)*(3)*(4)*(5)*(6)*(7)*(2*2*2)

(2)*(3)*(4)*(5)*(2*3)*(7)*(2*4)

(2)*(3)*(2*2)*(5)*(6)*(7)*(2*4)

(2)*(3)*(2*2)*(5)*(2*3)*(7)*(8)

(2)*(3)*(4)*(5)*(2*3)*(7)*(2*2*2)

(2)*(3)*(2*2)*(5)*(6)*(7)*(2*2*2)

(2)*(3)*(2*2)*(5)*(2*3)*(7)*(2*4)

(2)*(3)*(2*2)*(5)*(2*3)*(7)*(2*2*2)

MATHEMATICA

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

Table[Array[Length[facs[#]]&, n, 1, Times], {n, 30}]

CROSSREFS

Dominates A321468.

Cf. A001055, A050336, A066723, A076716, A157612, A281113, A300383, A317144, A317145, A321467, A321470, A321471, A321472.

Sequence in context: A330719 A095329 A327486 * A280306 A376511 A140723

Adjacent sequences: A321511 A321512 A321513 * A321515 A321516 A321517

KEYWORD

nonn

AUTHOR

Gus Wiseman, Nov 11 2018

STATUS

approved