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 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).
EXAMPLE
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).
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}]
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Sep 10 2018
EXTENSIONS
a(11)-a(12) from Robert Price, Sep 14 2018
STATUS
approved