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

Author

Gus Wiseman, Oct 17 2022

Keywords

• nonn

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.

Examples

			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)
		

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.

Programs

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}]