A317143 In the ranked poset of integer partitions ordered by refinement, row n lists the Heinz numbers of integer partitions finer (less) than or equal to the integer partition with Heinz number n.

1, 2, 3, 4, 4, 5, 6, 8, 6, 8, 7, 9, 10, 12, 16, 8, 9, 12, 16, 10, 12, 16, 11, 14, 15, 18, 20, 24, 32, 12, 16, 13, 21, 22, 25, 27, 28, 30, 36, 40, 48, 64, 14, 18, 20, 24, 32, 15, 18, 20, 24, 32, 16, 17, 26, 33, 35, 42, 44, 45, 50, 54, 56, 60, 72, 80, 96, 128

Offset

1,2

Comments

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

If x and y are partitions of the same integer and it is possible to produce x by further partitioning the parts of y, flattening, and sorting, then x <= y.

Links

Table of n, a(n) for n=1..68.

Example

The partitions finer than or equal to (2,2) are (2,2), (2,1,1), (1,1,1,1), with Heinz numbers 9, 12, 16, so the 9th row is {9, 12, 16}.
Triangle begins:
   1
   2
   3   4
   4
   5   6   8
   6   8
   7   9  10  12  16
   8
   9  12  16
  10  12  16
  11  14  15  18  20  24  32
  12  16
  13  21  22  25  27  28  30  36  40  48  64
  14  18  20  24  32
  15  18  20  24  32
  16
  17  26  33  35  42  44  45  50  54  56  60  72  80  96 128

Mathematica

primeMS[n_]:=If[n===1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Table[Union[Times@@@Map[Prime, Join@@@Tuples[IntegerPartitions/@primeMS[n]], {2}]], {n, 12}]

Crossrefs

Row lengths are A300383.

Cf. A002846, A056239, A213427, A215366, A265947, A296150, A299201, A317141.

Sequence in context: A017876 A356860 A017865 * A351520 A065328 A049877

Adjacent sequences: A317140 A317141 A317142 * A317144 A317145 A317146

Keyword

nonn,tabf

Author

Gus Wiseman, Jul 22 2018

Status

approved