A321471 Heinz numbers of integer partitions that can be partitioned into blocks with sums {1, 2, ..., k} for some 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

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.

Links

Table of n, a(n) for n=1..49.

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.

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

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

Sequence in context: A020696 A328769 A290249 * A216762 A132269 A053287

Adjacent sequences: A321468 A321469 A321470 * A321472 A321473 A321474

Keyword

nonn

Author

Gus Wiseman, Nov 13 2018

Status

approved