A330106 Number of integer partitions of n whose product is a powerful number.

0, 0, 0, 0, 2, 2, 5, 5, 9, 11, 18, 19, 30, 36, 51, 62, 87, 104, 141, 171, 225, 271, 349, 419, 534, 643, 804, 965, 1197, 1431, 1766, 2106, 2571, 3063, 3719, 4410, 5325, 6305, 7567, 8939, 10678, 12572, 14961, 17567, 20804, 24389, 28775, 33626, 39551, 46106

Offset: 0

Author

Gus Wiseman, Dec 05 2019

Keywords

• nonn

Examples

			The a(4) = 2 through a(10) = 18 partitions:
  (4)   (41)   (33)    (331)    (8)       (9)        (55)
  (22)  (221)  (42)    (421)    (44)      (81)       (82)
               (222)   (2221)   (422)     (333)      (91)
               (411)   (4111)   (2222)    (441)      (433)
               (2211)  (22111)  (3311)    (4221)     (442)
                                (4211)    (22221)    (811)
                                (22211)   (33111)    (3322)
                                (41111)   (42111)    (3331)
                                (221111)  (222111)   (4222)
                                          (411111)   (4411)
                                          (2211111)  (22222)
                                                     (42211)
                                                     (222211)
                                                     (331111)
                                                     (421111)
                                                     (2221111)
                                                     (4111111)
                                                     (22111111)
		

Crossrefs

The strict version is A330216.

Powerful numbers are A001694.

Partitions whose product is a perfect power are A320322.

Cf. A000041, A001055, A001597, A003963, A064573, A320325.

Programs

Mathematica

powQ[n_]:=Min@@Last/@FactorInteger[n]>1;
Table[Length[Select[IntegerPartitions[n],powQ[Times@@#]&]],{n,0,30}]