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A319909 Number of distinct positive 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 1^n.

%I A319909 #5 Oct 01 2018 21:16:59
%S A319909 0,1,1,1,1,2,4,5,10,15,21,34,49,68,101,142,197,280,387,538,751,1045,
%T A319909 1442,2010,2772,3865,5339,7396,10273,14201,19693
%N A319909 Number of distinct positive 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 1^n.
%e A319909 We have
%e A319909    7 = 1+1+1+1+1+1+1,
%e A319909    8 = (1+1)*(1+1+1)+1+1,
%e A319909    9 = (1+1)*(1+1)*(1+1)+1,
%e A319909   10 = (1+1+1+1+1)*(1+1),
%e A319909   12 = (1+1+1)*(1+1+1+1),
%e A319909 so a(7) = 5.
%t A319909 ReplaceListRepeated[forms_,rerules_]:=Union[Flatten[FixedPointList[Function[pre,Union[Flatten[ReplaceList[#,rerules]&/@pre,1]]],forms],1]];
%t A319909 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&]];
%t A319909 Table[Length[mexos[Table[1,{n}]]],{n,30}]
%Y A319909 Cf. A000792, A001970, A005520, A048249, A066739, A066815, A070960, A201163, A275870, A319850, A318948, A318949, A319912, A319913.
%K A319909 nonn,more
%O A319909 0,6
%A A319909 _Gus Wiseman_, Oct 01 2018