A330216 Number of strict integer partitions of n whose product is a powerful number.
0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2, 3, 4, 3, 4, 5, 7, 8, 8, 10, 12, 12, 15, 18, 19, 20, 24, 25, 28, 38, 41, 43, 50, 55, 63, 79, 85, 88, 104, 116, 124, 143, 157, 173, 197, 214, 235, 274, 294, 319, 363, 393, 430, 487, 529, 577, 647, 692, 752, 856, 925, 992, 1099
Offset: 0
Keywords
Examples
The a(n) partitions for n = 4, 9, 12, 13, 16, 17, 18:
(4) (9) (8,4) (9,4) (16) (9,8) (12,6)
(8,1) (9,3) (6,4,3) (9,4,3) (16,1) (16,2)
(6,3,2,1) (8,4,1) (12,3,1) (8,6,3) (9,8,1)
(9,3,1) (9,4,2,1) (9,6,2) (8,6,3,1)
(6,4,3,2,1) (10,5,2) (9,4,3,2)
(12,3,2) (9,6,2,1)
(9,4,3,1) (10,5,2,1)
(12,3,2,1)
Crossrefs
Programs
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Mathematica
powQ[n_]:=Min@@Last/@FactorInteger[n]>1; Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&powQ[Times@@#]&]],{n,0,30}]