A319911 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.

0, 1, 1, 2, 3, 7, 9, 21, 31, 65, 102, 194, 321, 575, 956, 1652, 2684, 4576, 7367, 12035, 19490, 31185, 49418, 78595, 123393

Offset: 1

Author

Gus Wiseman, Oct 01 2018

Keywords

• nonn
• more

Examples

			The a(6) = 7 pairs:
  6 <= (6)
  6 <= (4,2)
  8 <= (4,2)
  6 <= (3,3)
  9 <= (3,3)
  6 <= (2,2,2)
  8 <= (2,2,2)
The a(7) = 9 pairs:
   7 <= (7)
   7 <= (5,2)
  10 <= (5,2)
   7 <= (4,3)
  12 <= (4,3)
   7 <= (3,2,2)
   8 <= (3,2,2)
  10 <= (3,2,2)
  12 <= (3,2,2)
		

Crossrefs

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

Programs

Mathematica

ReplaceListRepeated[forms_,rerules_]:=Union[Flatten[FixedPointList[Function[pre,Union[Flatten[ReplaceList[#,rerules]&/@pre,1]]],forms],1]];
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&]];
Table[Total[Length/@nexos/@Select[IntegerPartitions[n],FreeQ[#,1]&]],{n,30}]