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A323088
Number of strict integer partitions of n using numbers that are not perfect powers.
4
1, 0, 1, 1, 0, 2, 1, 2, 2, 2, 3, 3, 4, 5, 5, 7, 7, 9, 11, 11, 15, 16, 18, 22, 24, 27, 32, 34, 41, 45, 51, 59, 64, 75, 82, 94, 105, 116, 132, 146, 163, 183, 202, 225, 251, 277, 309, 341, 378, 417, 463, 510, 564, 622, 685, 754, 830, 914, 1001, 1103, 1207, 1325
OFFSET
0,6
FORMULA
O.g.f.: Product_{n in A007916} (1 + x^n).
EXAMPLE
A list of all strict integer partitions using numbers that are not perfect powers begins:
2: (2) 11: (6,3,2) 15: (13,2) 17: (12,5)
3: (3) 12: (12) 15: (12,3) 17: (12,3,2)
5: (5) 12: (10,2) 15: (10,5) 17: (11,6)
5: (3,2) 12: (7,5) 15: (10,3,2) 17: (10,7)
6: (6) 12: (7,3,2) 15: (7,6,2) 17: (10,5,2)
7: (7) 13: (13) 15: (7,5,3) 17: (7,5,3,2)
7: (5,2) 13: (11,2) 16: (14,2) 18: (18)
8: (6,2) 13: (10,3) 16: (13,3) 18: (15,3)
8: (5,3) 13: (7,6) 16: (11,5) 18: (13,5)
9: (7,2) 13: (6,5,2) 16: (11,3,2) 18: (13,3,2)
9: (6,3) 14: (14) 16: (10,6) 18: (12,6)
10: (10) 14: (12,2) 16: (7,6,3) 18: (11,7)
10: (7,3) 14: (11,3) 16: (6,5,3,2) 18: (11,5,2)
10: (5,3,2) 14: (7,5,2) 17: (17) 18: (10,6,2)
11: (11) 14: (6,5,3) 17: (15,2) 18: (10,5,3)
11: (6,5) 15: (15) 17: (14,3) 18: (7,6,5)
MATHEMATICA
perpowQ[n_]:=GCD@@FactorInteger[n][[All, 2]]>1;
Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&FreeQ[#, 1]&&And@@Not/@perpowQ/@#&]], {n, 20}]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jan 04 2019
STATUS
approved