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

Offset: 1

Author

Gus Wiseman, Sep 11 2018

Keywords

• nonn

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.

Examples

			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).
		

Crossrefs

Cf. A000108, A001055, A007716, A074206, A118376, A277130, A281118, A281119, A292505, A317144, A319122, A319138.

Programs

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];
Table[Length[Select[otfs[n],FreeQ[#,_Integer?(!PrimeQ[#]&)]&]],{n,100}]

Formula

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

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