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.

1, 2, 3, 5, 12, 30, 53, 128, 247, 493, 989, 1889, 3434, 6390, 11526, 20400, 35818, 62083, 106223, 180170

Offset

1,2

Links

Table of n, a(n) for n=1..20.

Example

The a(6) = 30 pairs:
  1 <= (1)
  2 <= (2)
  2 <= (1,1)
  3 <= (3)
  3 <= (2,1)
  3 <= (1,1,1)
  4 <= (4)
  4 <= (2,2)
  4 <= (3,1)
  4 <= (2,1,1)
  4 <= (1,1,1,1)
  5 <= (5)
  5 <= (3,2)
  5 <= (4,1)
  5 <= (2,2,1)
  5 <= (3,1,1)
  5 <= (2,1,1,1)
  5 <= (1,1,1,1,1)
  6 <= (6)
  6 <= (3,2)
  6 <= (3,3)
  6 <= (4,2)
  6 <= (5,1)
  6 <= (2,2,1)
  6 <= (2,2,2)
  6 <= (3,1,1)
  6 <= (3,2,1)
  6 <= (4,1,1)
  6 <= (2,1,1,1)
  6 <= (2,2,1,1)
  6 <= (3,1,1,1)
  6 <= (1,1,1,1,1)
  6 <= (2,1,1,1,1)
  6 <= (1,1,1,1,1,1)

Mathematica

ReplaceListRepeated[forms_, rerules_]:=Union[Flatten[FixedPointList[Function[pre, Union[Flatten[ReplaceList[#, rerules]&/@pre, 1]]], forms], 1]];
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&]];
Table[Total[Length/@mexos/@IntegerPartitions[n]], {n, 20}]

Crossrefs

Cf. A000792, A001970, A005520, A048249, A066739, A066815, A275870, A319850, A318949, A319909, A319910, A319911, A319913.

Sequence in context: A331811 A284910 A233231 * A153987 A228905 A369847

Adjacent sequences: A319909 A319910 A319911 * A319913 A319914 A319915

Keyword

nonn,more

Author

Gus Wiseman, Oct 01 2018

Status

approved