proposed

approved

editing

proposed

Cf. A321467, A321468, A321470, A321471, A321472, A321514.

allocated Heinz numbers of integer partitions that can be partitioned into blocks with sums {1, 2, ..., k} for Gus Wisemansome k.

2, 6, 8, 30, 36, 40, 48, 64, 210, 252, 270, 280, 300, 324, 336, 360, 400, 432, 448, 480, 576, 640, 768, 1024, 2310, 2772, 2940, 2970, 3080, 3150, 3300, 3528, 3564, 3696, 3780, 3920, 3960, 4050, 4200, 4400, 4500, 4536, 4704, 4752, 4860, 4928, 5040, 5280, 5400

1,1

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.

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.

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@@#]&]!={}&]

Subsequence of A242422.

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

Cf. A321467, A321468, A321470, A321471, A321472, A321514.

allocated

nonn

Gus Wiseman, Nov 13 2018

approved

editing

allocated for Gus Wiseman

allocated

approved