A319119 Number of multimin tree-factorizations of Heinz numbers of integer partitions of n.

1, 3, 9, 37, 173, 921, 5185, 30497, 184469, 1140413, 7170085, 45704821

Offset: 1

Author

Gus Wiseman, Sep 10 2018

Keywords

• nonn
• more

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.

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

Examples

			The a(3) = 9 multimin tree-factorizations:
  5, 6, 8,
  (2*3), (2*4), (4*2), (2*2*2),
  (2*(2*2)), ((2*2)*2).
Or as series-reduced plane trees of multisets:
  3, 12, 111,
  (1,2), (1,11), (11,1), (1,1,1),
  (1,(1,1)), ((1,1),1).
		

Crossrefs

Cf. A001055, A020639, A196545, A255397, A281113, A281118, A281119, A295279, A317545, A317546, A319118.

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[Sum[Length[mmftrees[k]],{k,Times@@Prime/@#&/@IntegerPartitions[n]}],{n,7}]

Extensions

a(11)-a(12) from Robert Price, Sep 14 2018