A319841 Number of distinct positive integers that can be obtained by iteratively adding or multiplying together parts of the integer partition with Heinz number n until only one part remains.

0, 1, 1, 2, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 2, 4, 1, 3, 1, 4, 2, 2, 1, 5, 2, 2, 2, 4, 1, 5, 1, 6, 2, 2, 2, 6, 1, 2, 2, 7, 1, 6, 1, 4, 4, 2, 1, 8, 2, 5, 2, 4, 1, 6, 2, 8, 2, 2, 1, 7, 1, 2, 4, 9, 2, 6, 1, 4, 2, 6, 1, 8, 1, 2, 6, 4, 2, 6, 1, 9, 4, 2, 1, 10, 2, 2, 2

Offset

1,4

Links

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

Formula

a(2^n) = A048249(n).

Example

60 is the Heinz number of (3,2,1,1) and
   5 = (3+2)*1*1
   6 = 3*2*1*1
   7 = 3+2+1+1
   8 = (3+1)*2*1
   9 = 3*(2+1)*1
  10 = (3+2)*(1+1)
  12 = (3+1)*(2+1)
so we have a(60) = 7. It is not possible to obtain 11 by adding or multiplying together the parts of (3,2,1,1).

Mathematica

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

Crossrefs

Cf. A001055, A001970, A048249, A056239, A063834, A066739, A066815, A281113, A318948, A318949, A319855, A319856.

Sequence in context: A091090 A305052 A305079 * A336099 A290090 A066075

Adjacent sequences: A319838 A319839 A319840 * A319842 A319843 A319844

Keyword

nonn

Author

Gus Wiseman, Sep 29 2018

Status

approved