A319911 - OEIS

%I #5 Oct 01 2018 21:17:15

%S 0,1,1,2,3,7,9,21,31,65,102,194,321,575,956,1652,2684,4576,7367,12035,

%T 19490,31185,49418,78595,123393

%N Number of distinct pairs (m, y), where m >= 1 and y is an integer partition of n with no 1's, 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(6) = 7 pairs:

%e   6 <= (6)

%e   6 <= (4,2)

%e   8 <= (4,2)

%e   6 <= (3,3)

%e   9 <= (3,3)

%e   6 <= (2,2,2)

%e   8 <= (2,2,2)

%e The a(7) = 9 pairs:

%e    7 <= (7)

%e    7 <= (5,2)

%e   10 <= (5,2)

%e    7 <= (4,3)

%e   12 <= (4,3)

%e    7 <= (3,2,2)

%e    8 <= (3,2,2)

%e   10 <= (3,2,2)

%e   12 <= (3,2,2)

%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/@Select[IntegerPartitions[n],FreeQ[#,1]&]],{n,30}]

%Y Cf. A000792, A001970, A005520, A048249, A066739, A066815, A275870, A319850, A318949, A319909, A319910, A319912, A319913.

%K nonn,more

%O 1,4

%A _Gus Wiseman_, Oct 01 2018