This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A319910 #5 Oct 01 2018 21:17:09
%S A319910 1,3,6,11,23,48,85,178,331,619,1176,2183,3876,7013,12447,21719,37628,
%T A319910 64570,109723,185055
%N A319910 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 A319910 The a(4) = 11 pairs:
%e A319910 4 <= (4)
%e A319910 3 <= (3,1)
%e A319910 4 <= (3,1)
%e A319910 4 <= (2,2)
%e A319910 2 <= (2,1,1)
%e A319910 3 <= (2,1,1)
%e A319910 4 <= (2,1,1)
%e A319910 1 <= (1,1,1,1)
%e A319910 2 <= (1,1,1,1)
%e A319910 3 <= (1,1,1,1)
%e A319910 4 <= (1,1,1,1)
%t A319910 ReplaceListRepeated[forms_,rerules_]:=Union[Flatten[FixedPointList[Function[pre,Union[Flatten[ReplaceList[#,rerules]&/@pre,1]]],forms],1]];
%t A319910 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 A319910 Table[Total[Length/@nexos/@IntegerPartitions[n]],{n,20}]
%Y A319910 Cf. A000792, A001970, A005520, A048249, A066739, A066815, A275870, A319850, A318949, A319909, A319912, A319913.
%K A319910 nonn,more
%O A319910 1,2
%A A319910 _Gus Wiseman_, Oct 01 2018