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

1,4

Links

Table of n, a(n) for n=1..25.

Example

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)

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}]

Crossrefs

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

Sequence in context: A225202 A046152 A057239 * A024541 A250267 A333517

Adjacent sequences: A319908 A319909 A319910 * A319912 A319913 A319914

Keyword

nonn,more

Author

Gus Wiseman, Oct 01 2018

Status

approved