A357858 Number of integer partitions that can be obtained by iteratively adding and multiplying together parts of the integer partition with Heinz number n.

1, 1, 1, 3, 1, 3, 1, 6, 2, 3, 1, 7, 1, 3, 3, 11, 1, 7, 1, 8, 3, 3, 1, 14, 3, 3, 4, 8, 1, 11, 1, 19, 3, 3, 3, 18, 1, 3, 3, 18, 1, 12, 1, 8, 8, 3, 1, 27, 3, 10, 3, 8, 1, 16, 3, 19, 3, 3, 1, 25, 1, 3, 8, 33, 3, 12, 1, 8, 3, 12, 1, 35, 1, 3, 11, 8, 3, 12, 1, 34, 9

Offset

1,4

Comments

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

Links

Table of n, a(n) for n=1..81.

Example

The a(n) partitions for n = 1, 4, 8, 9, 12, 16, 20, 24:
  ()  (1)   (1)    (4)   (2)    (1)     (3)    (2)
      (2)   (2)    (22)  (3)    (2)     (4)    (3)
      (11)  (3)          (4)    (3)     (5)    (4)
            (11)         (21)   (4)     (6)    (5)
            (21)         (22)   (11)    (31)   (6)
            (111)        (31)   (21)    (32)   (21)
                         (211)  (22)    (41)   (22)
                                (31)    (311)  (31)
                                (111)          (32)
                                (211)          (41)
                                (1111)         (211)
                                               (221)
                                               (311)
                                               (2111)

Mathematica

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
ReplaceListRepeated[forms_, rerules_]:=Union[Flatten[FixedPointList[Function[pre, Union[Flatten[ReplaceList[#, rerules]&/@pre, 1]]], forms], 1]];
Table[Length[ReplaceListRepeated[{primeMS[n]}, {{foe___, x_, mie___, y_, afe___}:>Sort[Append[{foe, mie, afe}, x+y]], {foe___, x_, mie___, y_, afe___}:>Sort[Append[{foe, mie, afe}, x*y]]}]], {n, 100}]

Crossrefs

The single-part partitions are counted by A319841, with an inverse A319913.

The minimum is A319855, maximum A319856.

A000041 counts integer partitions.

A001222 counts prime indices, distinct A001221.

A056239 adds up prime indices.

A066739 counts representations as a sum of products.

Cf. A000792, A001055, A001970, A005520, A048249, A063834, A066815, A318948, A319850, A319909, A319910.

Sequence in context: A236800 A367628 A126212 * A066637 A317144 A050336

Adjacent sequences: A357855 A357856 A357857 * A357859 A357860 A357861

Keyword

nonn

Author

Gus Wiseman, Oct 17 2022

Status

approved