A323089 Number of strict integer partitions of n using 1 and numbers that are not perfect powers.
1, 1, 1, 2, 1, 2, 3, 3, 4, 4, 5, 6, 7, 9, 10, 12, 14, 16, 20, 22, 26, 31, 34, 40, 46, 51, 59, 66, 75, 86, 96, 110, 123, 139, 157, 176, 199, 221, 248, 278, 309, 346, 385, 427, 476, 528, 586, 650, 719, 795, 880, 973, 1074, 1186, 1307, 1439, 1584, 1744, 1915, 2104
Offset: 0
Keywords
Examples
A list of all strict integer partitions using 1 and numbers that are not perfect powers begins: 1: (1) 8: (5,2,1) 12: (12) 14: (14) 2: (2) 9: (7,2) 12: (11,1) 14: (13,1) 3: (3) 9: (6,3) 12: (10,2) 14: (12,2) 3: (2,1) 9: (6,2,1) 12: (7,5) 14: (11,3) 4: (3,1) 9: (5,3,1) 12: (7,3,2) 14: (11,2,1) 5: (5) 10: (10) 12: (6,5,1) 14: (10,3,1) 5: (3,2) 10: (7,3) 12: (6,3,2,1) 14: (7,6,1) 6: (6) 10: (7,2,1) 13: (13) 14: (7,5,2) 6: (5,1) 10: (6,3,1) 13: (12,1) 14: (6,5,3) 6: (3,2,1) 10: (5,3,2) 13: (11,2) 14: (6,5,2,1) 7: (7) 11: (11) 13: (10,3) 15: (15) 7: (6,1) 11: (10,1) 13: (10,2,1) 15: (14,1) 7: (5,2) 11: (7,3,1) 13: (7,6) 15: (13,2) 8: (7,1) 11: (6,5) 13: (7,5,1) 15: (12,3) 8: (6,2) 11: (6,3,2) 13: (7,3,2,1) 15: (12,2,1) 8: (5,3) 11: (5,3,2,1) 13: (6,5,2) 15: (11,3,1)
Programs
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Mathematica
perpowQ[n_]:=GCD@@FactorInteger[n][[All,2]]>1; Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&And@@Not/@perpowQ/@#&]],{n,65}]
Formula
O.g.f.: (1 + x) * Product_{n in A007916} (1 + x^n).