proposed

approved

editing

proposed

Cf. A001055, A001970, A048249, A056239, A063834, A065026, A066739, A066815, A121229, A281113, A284639, A318948, A318949, A319855, A319856.

allocated for Gus WisemanNumber 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

1,4

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

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).

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}]

Cf. A001055, A001970, A048249, A063834, A065026, A066739, A066815, A121229, A281113, A284639, A318948, A318949.

allocated

nonn

Gus Wiseman, Sep 29 2018

approved

editing

allocated for Gus Wiseman

allocated

approved