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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.
4
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
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}]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Sep 29 2018
STATUS
approved