A319118 Number of multimin tree-factorizations of n.

1, 1, 1, 2, 1, 2, 1, 6, 2, 2, 1, 8, 1, 2, 2, 24, 1, 6, 1, 8, 2, 2, 1, 42, 2, 2, 6, 8, 1, 8, 1, 112, 2, 2, 2, 38, 1, 2, 2, 42, 1, 8, 1, 8, 8, 2, 1, 244, 2, 6, 2, 8, 1, 24, 2, 42, 2, 2, 1, 58, 1, 2, 8, 568, 2, 8, 1, 8, 2, 8, 1, 268, 1, 2, 6, 8, 2, 8, 1, 244, 24

Offset: 1

Author

Gus Wiseman, Sep 10 2018

Keywords

• nonn

Comments

A multimin factorization of n is an ordered factorization of n into factors greater than 1 such that the sequence of minimal primes dividing each factor is weakly increasing. A multimin tree-factorization of n is either the number n itself or a sequence of multimin tree-factorizations, one of each factor in a multimin factorization of n with at least two factors.

Examples

			The a(12) = 8 multimin tree-factorizations:
  12,
  (2*6), (4*3), (6*2), (2*2*3),
  (2*(2*3)), ((2*2)*3), ((2*3)*2).
Or as series-reduced plane trees of multisets:
  112,
  (1,12), (11,2), (12,1), (1,1,2),
  (1,(1,2)), ((1,1),2), ((1,2),1).
		

Crossrefs

Cf. A001055, A005804, A020639, A118376, A196545, A255397, A281113, A281118, A281119, A295279, A317545, A317546, A318577, A319119.

Programs

Mathematica

facs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#1,d]&)/@Select[facs[n/d],Min@@#1>=d&],{d,Rest[Divisors[n]]}]];
mmftrees[n_]:=Prepend[Join@@(Tuples[mmftrees/@#]&/@Select[Join@@Permutations/@Select[facs[n],Length[#]>1&],OrderedQ[FactorInteger[#][[1,1]]&/@#]&]),n];
Table[Length[mmftrees[n]],{n,100}]

Formula

a(prime^n) = A118376(n).

a(product of n distinct primes) = A005804(n).