A319136 - OEIS

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A319136

Number of complete planar branching factorizations of n.

0, 1, 1, 1, 1, 2, 1, 3, 1, 2, 1, 9, 1, 2, 2, 11, 1, 9, 1, 9, 2, 2, 1, 44, 1, 2, 3, 9, 1, 18, 1, 45, 2, 2, 2, 66, 1, 2, 2, 44, 1, 18, 1, 9, 9, 2, 1, 225, 1, 9, 2, 9, 1, 44, 2, 44, 2, 2, 1, 132, 1, 2, 9, 197, 2, 18, 1, 9, 2, 18, 1, 450, 1, 2, 9, 9, 2, 18, 1, 225

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OFFSET

1,6

COMMENTS

A planar branching factorization of n is either the number n itself or a sequence of at least two planar branching factorizations, one of each factor in an ordered factorization of n. A planar branching factorization is complete if the leaves are all prime numbers.

LINKS

Table of n, a(n) for n=1..80.

FORMULA

a(prime^n) = A001003(n - 1).

a(product of n distinct primes) = A032037(n).

EXAMPLE

The a(12) = 9 trees:

(2*2*3), (2*3*2), (3*2*2),

(2*(2*3)), (2*(3*2)), (3*(2*2)), ((2*2)*3), ((2*3)*2), ((3*2)*2).

MATHEMATICA

ordfacs[n_]:=If[n<=1, {{}}, Join@@Table[(Prepend[#1, d]&)/@ordfacs[n/d], {d, Rest[Divisors[n]]}]]

otfs[n_]:=Prepend[Join@@Table[Tuples[otfs/@f], {f, Select[ordfacs[n], Length[#]>1&]}], n];