OFFSET
1,1
COMMENTS
The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
These partitions are those that are finer than (k, ..., 3, 2, 1) in the poset of integer partitions of 1 + 2 + ... + k, for some k, ordered by refinement.
EXAMPLE
The sequence of all integer partitions whose Heinz numbers are in the sequence begins: (1), (21), (111), (321), (2211), (3111), (21111), (111111), (4321), (42211), (32221), (43111), (33211), (222211), (421111), (322111), (331111), (2221111), (4111111), (3211111), (22111111), (31111111), (211111111), (1111111111).
The partition (322111) has Heinz number 360 and can be partitioned as ((1)(2)(3)(112)), ((1)(2)(12)(13)), or ((1)(11)(3)(22)), so 360 belongs to the sequence.
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
Select[Range[2, 1000], Select[Map[Total[primeMS[#]]&, facs[#], {2}], Sort[#]==Range[Max@@#]&]!={}&]
CROSSREFS
Subsequence of A242422.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Nov 13 2018
STATUS
approved