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A319912 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 any two or multiplying any two non-1 parts of y until only one part (equal to m) remains.

%I A319912 #5 Oct 01 2018 21:17:25
%S A319912 1,2,3,5,12,30,53,128,247,493,989,1889,3434,6390,11526,20400,35818,
%T A319912 62083,106223,180170
%N A319912 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 any two or multiplying any two non-1 parts of y until only one part (equal to m) remains.
%e A319912 The a(6) = 30 pairs:
%e A319912   1 <= (1)
%e A319912   2 <= (2)
%e A319912   2 <= (1,1)
%e A319912   3 <= (3)
%e A319912   3 <= (2,1)
%e A319912   3 <= (1,1,1)
%e A319912   4 <= (4)
%e A319912   4 <= (2,2)
%e A319912   4 <= (3,1)
%e A319912   4 <= (2,1,1)
%e A319912   4 <= (1,1,1,1)
%e A319912   5 <= (5)
%e A319912   5 <= (3,2)
%e A319912   5 <= (4,1)
%e A319912   5 <= (2,2,1)
%e A319912   5 <= (3,1,1)
%e A319912   5 <= (2,1,1,1)
%e A319912   5 <= (1,1,1,1,1)
%e A319912   6 <= (6)
%e A319912   6 <= (3,2)
%e A319912   6 <= (3,3)
%e A319912   6 <= (4,2)
%e A319912   6 <= (5,1)
%e A319912   6 <= (2,2,1)
%e A319912   6 <= (2,2,2)
%e A319912   6 <= (3,1,1)
%e A319912   6 <= (3,2,1)
%e A319912   6 <= (4,1,1)
%e A319912   6 <= (2,1,1,1)
%e A319912   6 <= (2,2,1,1)
%e A319912   6 <= (3,1,1,1)
%e A319912   6 <= (1,1,1,1,1)
%e A319912   6 <= (2,1,1,1,1)
%e A319912   6 <= (1,1,1,1,1,1)
%t A319912 ReplaceListRepeated[forms_,rerules_]:=Union[Flatten[FixedPointList[Function[pre,Union[Flatten[ReplaceList[#,rerules]&/@pre,1]]],forms],1]];
%t A319912 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 A319912 Table[Total[Length/@mexos/@IntegerPartitions[n]],{n,20}]
%Y A319912 Cf. A000792, A001970, A005520, A048249, A066739, A066815, A275870, A319850, A318949, A319909, A319910, A319911, A319913.
%K A319912 nonn,more
%O A319912 1,2
%A A319912 _Gus Wiseman_, Oct 01 2018