%I #9 Oct 01 2018 21:16:51
%S 1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,2,1,2,2,1,1,2,2,1,2,2,1,4,1,2,2,1,
%T 2,4,1,1,2,4,1,4,1,2,4,1,1,4,2,3,2,2,1,5,2,4,2,1,1,5,1,1,4,4,2,4,1,2,
%U 2,4,1,5,1,1,6,2,2,4,1,5,4,1,1,7,2,1,2
%N Number of distinct 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 the integer partition with Heinz number n.
%C The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
%e The Heinz number of (3,3,2) is 75 and we have
%e 3+3+2 = 8,
%e 3+3*2 = 9,
%e 3*3+2 = 11,
%e (3+3)*2 = 12,
%e 3*(3+2) = 15,
%e 3*3*2 = 18,
%e so a(75) = 6.
%t ReplaceListRepeated[forms_,rerules_]:=Union[Flatten[FixedPointList[Function[pre,Union[Flatten[ReplaceList[#,rerules]&/@pre,1]]],forms],1]];
%t 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 Table[Length[mexos[If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]]],{n,100}]
%Y Cf. A000792, A001970, A005520, A048249, A066739, A070960, A201163, A275870, A319850, A318949, A319855, A319856, A319909, A319912, A319913.
%K nonn
%O 1,15
%A _Gus Wiseman_, Oct 01 2018