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

0,5

Links

Table of n, a(n) for n=0..49.

Example

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)

Mathematica

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

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.

Sequence in context: A213675 A321185 A320322 * A214795 A287744 A214253

Adjacent sequences: A330103 A330104 A330105 * A330107 A330108 A330109

Keyword

nonn

Author

Gus Wiseman, Dec 05 2019

Status

approved