%I #5 Nov 14 2018 08:09:27

%S 2,5,6,13,21,22,25,29,30,46,47,57,73,85,86,91,102,107,121,123,130,142,

%T 147,151,154,165,175,185,197,201,206,210,217,222,257,298,299

%N Heinz numbers of integer partitions whose parts can be further partitioned and flattened to obtain the partition (k, ..., 3, 2, 1) for some k.

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

%C These partitions are those that are coarser than (k, ..., 3, 2, 1) in the poset of integer partitions of 1 + 2 + ... + k, for some k, ordered by refinement.

%e The sequence of all integer partitions whose Heinz numbers are in the sequence begins: (1), (3), (2,1), (6), (4,2), (5,1), (3,3), (10), (3,2,1), (9,1), (15), (8,2), (21), (7,3), (14,1), (6,4), (7,2,1), (28), (5,5), (13,2), (6,3,1), (20,1), (4,4,2), (36), (5,4,1), (5,3,2), (4,3,3), (12,3), (45), (19,2), (27,1), (4,3,2,1).

%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t Select[Range[2,200],Select[Sort/@Join@@@Tuples[IntegerPartitions/@primeMS[#]],Sort[#]==Range[Max@@#]&]!={}&]

%Y Subsequence of A242422.

%Y Cf. A001970, A002846, A056239, A066723, A112798, A213427, A242422, A265947, A300383, A317141.

%Y Cf. A321467, A321468, A321470, A321471, A321514.

%K nonn

%O 1,1

%A _Gus Wiseman_, Nov 13 2018