%I #5 Oct 01 2018 21:17:09

%S 1,3,6,11,23,48,85,178,331,619,1176,2183,3876,7013,12447,21719,37628,

%T 64570,109723,185055

%N Number of distinct pairs (m, y), where m >= 1 and y is an integer partition of n, such that m can be obtained by iteratively adding or multiplying together parts of y until only one part (equal to m) remains.

%e The a(4) = 11 pairs:

%e 4 <= (4)

%e 3 <= (3,1)

%e 4 <= (3,1)

%e 4 <= (2,2)

%e 2 <= (2,1,1)

%e 3 <= (2,1,1)

%e 4 <= (2,1,1)

%e 1 <= (1,1,1,1)

%e 2 <= (1,1,1,1)

%e 3 <= (1,1,1,1)

%e 4 <= (1,1,1,1)

%t ReplaceListRepeated[forms_,rerules_]:=Union[Flatten[FixedPointList[Function[pre,Union[Flatten[ReplaceList[#,rerules]&/@pre,1]]],forms],1]];

%t nexos[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_,mie___,y_,afe___}:>Sort[Append[{foe,mie,afe},x*y]]}],Length[#]==1&]];

%t Table[Total[Length/@nexos/@IntegerPartitions[n]],{n,20}]

%Y Cf. A000792, A001970, A005520, A048249, A066739, A066815, A275870, A319850, A318949, A319909, A319912, A319913.

%K nonn,more

%O 1,2

%A _Gus Wiseman_, Oct 01 2018