login
A319121
Number of complete multimin tree-factorizations of Heinz numbers of integer partitions of n.
1
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).
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}]
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Sep 11 2018
EXTENSIONS
a(11)-a(12) from Robert Price, Sep 14 2018
STATUS
approved