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

0,10

Links

Table of n, a(n) for n=0..63.

Example

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)

Mathematica

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

Crossrefs

The non-strict version is A330106.

Powerful numbers are A001694.

Partitions whose product is a perfect power are A320322.

Cf. A000009, A045778, A001597, A003963, A064573, A320325.

Sequence in context: A238957 A238970 A085755 * A241952 A236919 A138304

Adjacent sequences: A330213 A330214 A330215 * A330217 A330218 A330219

Keyword

nonn

Author

Gus Wiseman, Dec 05 2019

Status

approved