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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.
2
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
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}]
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Oct 01 2018
STATUS
approved