A319121 Number of complete multimin tree-factorizations of Heinz numbers of integer partitions of n.

1, 2, 5, 18, 74, 344, 1679, 8548, 44690, 238691, 1295990, 7132509

Offset

1,2

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 at least two multimin tree-factorizations, one of each factor in a multimin factorization of n. A multimin tree-factorization is complete if the leaves are all prime numbers.

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

Links

Table of n, a(n) for n=1..12.

Example

The a(3) = 5 trees are: 5, (2*3), (2*2*2), (2*(2*2)), ((2*2)*2).
The a(4) = 18 trees (normalized with prime(n) -> n):
  4,
  (13), (22), (112), (1111),
  (1(12)), ((12)1), ((11)2),
  (11(11)), (1(11)1), ((11)11), (1(111)), ((111)1), ((11)(11)),
  (1(1(11))), (1((11)1)), ((1(11))1), (((11)1)1).

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[Select[mmftrees[k], FreeQ[#, _Integer?(!PrimeQ[#]&)]&]], {k, Times@@Prime/@#&/@IntegerPartitions[n]}], {n, 10}]

Crossrefs

Cf. A000311, A001003, A001055, A020639, A255397, A281113, A281118, A281119, A295281, A317545, A317546, A318577, A319118, A319119.

Sequence in context: A039744 A344262 A352985 * A289655 A189281 A006848

Adjacent sequences: A319118 A319119 A319120 * A319122 A319123 A319124

Keyword

nonn,more

Author

Gus Wiseman, Sep 11 2018

Extensions

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

Status

approved