A319137 Number of strict planar branching factorizations of n.

1, 1, 1, 1, 1, 3, 1, 3, 1, 3, 1, 9, 1, 3, 3, 7, 1, 9, 1, 9, 3, 3, 1, 37, 1, 3, 3, 9, 1, 25, 1, 21, 3, 3, 3, 57, 1, 3, 3, 37, 1, 25, 1, 9, 9, 3, 1, 161, 1, 9, 3, 9, 1, 37, 3, 37, 3, 3, 1, 153, 1, 3, 9, 75, 3, 25, 1, 9, 3, 25, 1, 345, 1, 3, 9, 9, 3, 25, 1, 161

Offset: 1

Author

Gus Wiseman, Sep 11 2018

Keywords

• nonn

Comments

A strict planar branching factorization of n is either the number n itself or a sequence of at least two strict planar branching factorizations, one of each factor in a strict ordered factorization of n.

Examples

			The a(12) = 9 trees:
  12,
  (2*6), (3*4), (4*3),(6*2),
  (2*(2*3)), (2*(3*2)), ((2*3)*2), ((3*2)*2).
		

Crossrefs

Cf. A000108, A001055, A045778, A074206, A118376, A277130, A281113, A281118, A292504, A295279, A295281, A317144, A319122, A319123, A319138.

Programs

Mathematica

ordfacs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#1,d]&)/@ordfacs[n/d],{d,Rest[Divisors[n]]}]]
sotfs[n_]:=Prepend[Join@@Table[Tuples[sotfs/@f],{f,Select[ordfacs[n],And[Length[#]>1,UnsameQ@@#]&]}],n];
Table[Length[sotfs[n]],{n,100}]

Formula

a(prime^n) = A319123(n + 1).

a(product of n distinct primes) = A319122(n).