A319071 Number of integer partitions of n whose product of parts is a perfect power and whose parts all have the same number of prime factors, counted with multiplicity.

1, 0, 0, 0, 2, 0, 2, 0, 3, 2, 3, 0, 4, 1, 4, 3, 7, 1, 7, 1, 8, 6, 8, 0, 15, 5, 12, 6, 15, 4, 22, 4, 24, 12, 22, 8, 35, 7, 30, 16, 42, 9, 50, 9, 50, 30, 53, 7, 79, 22, 72, 33, 87, 21, 109, 26, 111, 55, 117, 24, 168, 40, 149, 65, 178, 59

Offset: 0

Author

Gus Wiseman, Oct 10 2018

Keywords

• nonn

Comments

The positions of zeros appear to be A048278.

Examples

			The a(4) = 2 through a(16) = 7 integer partitions (G = 16):
  4   33   8     9    55     66      94  77       555     G
  22  222  44    333  3322   444         5522     33333   88
           2222       22222  3333        332222   333222  664
                             222222      2222222          4444
                                                          5533
                                                          333322
                                                          22222222
		

Crossrefs

Cf. A003963, A048278, A064573, A279787, A305551, A319056, A319066, A319169, A320322, A320323.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],And[GCD@@FactorInteger[Times@@#][[All,2]]>1,SameQ@@PrimeOmega/@#]&]],{n,30}]