Voozh

A144757

Number of factor trees for n.

1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 6, 1, 2, 2, 5, 1, 6, 1, 6, 2, 2, 1, 20, 1, 2, 2, 6, 1, 12, 1, 14, 2, 2, 2, 30, 1, 2, 2, 20, 1, 12, 1, 6, 6, 2, 1, 70, 1, 6, 2, 6, 1, 20, 2, 20, 2, 2, 1, 60, 1, 2, 6, 42, 2, 12, 1, 6, 2, 12, 1, 140, 1, 2, 6, 6, 2, 12, 1, 70, 5, 2, 1, 60, 2, 2, 2, 20, 1, 60, 2, 6, 2, 2, 2, 252

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OFFSET

2,5

COMMENTS

A factor tree for n is a binary tree, with the root labeled with n and the terminal nodes labeled with primes, such that each non-terminal node is the product of its two child nodes. This is the number of prime factorizations of n, ignoring the commutativity and associativity of multiplication.

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 2..10000

FORMULA

a(n) = A000108(A001222(n)-1) * A008480(n).

EXAMPLE

a(12)=6 because 12 can be factored as (2*2)*3, (2*3)*2, (3*2)*2, 2*(2*3), 2*(3*2) and 3*(2*2).

MATHEMATICA

a8480[n_] := With[{f = FactorInteger[n][[All, 2]]}, Total[f]!/Times @@ (f!)];

a[n_] := CatalanNumber[PrimeOmega[n] - 1] * a8480[n];

a /@ Range[2, 100] (* Jean-François Alcover, Sep 20 2019 *)

PROG

(Haskell)