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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.

%I A319841 #5 Sep 30 2018 20:26:48
%S A319841 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,
%T A319841 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,
%U A319841 2,6,1,8,1,2,6,4,2,6,1,9,4,2,1,10,2,2,2
%N 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.
%F A319841 a(2^n) = A048249(n).
%e A319841 60 is the Heinz number of (3,2,1,1) and
%e A319841    5 = (3+2)*1*1
%e A319841    6 = 3*2*1*1
%e A319841    7 = 3+2+1+1
%e A319841    8 = (3+1)*2*1
%e A319841    9 = 3*(2+1)*1
%e A319841   10 = (3+2)*(1+1)
%e A319841   12 = (3+1)*(2+1)
%e A319841 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).
%t A319841 ReplaceListRepeated[forms_,rerules_]:=Union[Flatten[FixedPointList[Function[pre,Union[Flatten[ReplaceList[#,rerules]&/@pre,1]]],forms],1]];
%t A319841 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}]
%Y A319841 Cf. A001055, A001970, A048249, A056239, A063834, A066739, A066815, A281113, A318948, A318949, A319855, A319856.
%K A319841 nonn
%O A319841 1,4
%A A319841 _Gus Wiseman_, Sep 29 2018