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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.
7

%I #21 Oct 11 2018 10:09:41

%S 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,

%T 24,12,22,8,35,7,30,16,42,9,50,9,50,30,53,7,79,22,72,33,87,21,109,26,

%U 111,55,117,24,168,40,149,65,178,59

%N 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.

%C The positions of zeros appear to be A048278.

%e The a(4) = 2 through a(16) = 7 integer partitions (G = 16):

%e 4 33 8 9 55 66 94 77 555 G

%e 22 222 44 333 3322 444 5522 33333 88

%e 2222 22222 3333 332222 333222 664

%e 222222 2222222 4444

%e 5533

%e 333322

%e 22222222

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

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

%K nonn

%O 0,5

%A _Gus Wiseman_, Oct 10 2018