A319907 Number of distinct integers that can be obtained by iteratively adding any two or multiplying any two non-1 parts of an integer partition until only one part remains, starting with the integer partition with Heinz number n.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 1, 2, 2, 1, 2, 2, 1, 4, 1, 2, 2, 1, 2, 4, 1, 1, 2, 4, 1, 4, 1, 2, 4, 1, 1, 4, 2, 3, 2, 2, 1, 5, 2, 4, 2, 1, 1, 5, 1, 1, 4, 4, 2, 4, 1, 2, 2, 4, 1, 5, 1, 1, 6, 2, 2, 4, 1, 5, 4, 1, 1, 7, 2, 1, 2

Offset

1,15

Comments

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

Links

Table of n, a(n) for n=1..87.

Example

The Heinz number of (3,3,2) is 75 and we have
    3+3+2 = 8,
    3+3*2 = 9,
    3*3+2 = 11,
  (3+3)*2 = 12,
  3*(3+2) = 15,
    3*3*2 = 18,
so a(75) = 6.

Mathematica

ReplaceListRepeated[forms_, rerules_]:=Union[Flatten[FixedPointList[Function[pre, Union[Flatten[ReplaceList[#, rerules]&/@pre, 1]]], forms], 1]];
mexos[ptn_]:=If[Length[ptn]==0, {0}, Union@@Select[ReplaceListRepeated[{Sort[ptn]}, {{foe___, x_, mie___, y_, afe___}:>Sort[Append[{foe, mie, afe}, x+y]], {foe___, x_?(#>1&), mie___, y_?(#>1&), afe___}:>Sort[Append[{foe, mie, afe}, x*y]]}], Length[#]==1&]];
Table[Length[mexos[If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]]]], {n, 100}]

Crossrefs

Cf. A000792, A001970, A005520, A048249, A066739, A070960, A201163, A275870, A319850, A318949, A319855, A319856, A319909, A319912, A319913.

Sequence in context: A079706 A250005 A369179 * A357112 A078703 A090629

Adjacent sequences: A319904 A319905 A319906 * A319908 A319909 A319910

Keyword

nonn

Author

Gus Wiseman, Oct 01 2018

Status

approved