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%I #9 Mar 06 2019 08:52:10
%S 1,0,1,1,0,2,1,2,2,2,3,3,4,5,5,7,7,9,11,11,15,16,18,22,24,27,32,34,41,
%T 45,51,59,64,75,82,94,105,116,132,146,163,183,202,225,251,277,309,341,
%U 378,417,463,510,564,622,685,754,830,914,1001,1103,1207,1325
%N Number of strict integer partitions of n using numbers that are not perfect powers.
%F O.g.f.: Product_{n in A007916} (1 + x^n).
%e A list of all strict integer partitions using numbers that are not perfect powers begins:
%e 2: (2) 11: (6,3,2) 15: (13,2) 17: (12,5)
%e 3: (3) 12: (12) 15: (12,3) 17: (12,3,2)
%e 5: (5) 12: (10,2) 15: (10,5) 17: (11,6)
%e 5: (3,2) 12: (7,5) 15: (10,3,2) 17: (10,7)
%e 6: (6) 12: (7,3,2) 15: (7,6,2) 17: (10,5,2)
%e 7: (7) 13: (13) 15: (7,5,3) 17: (7,5,3,2)
%e 7: (5,2) 13: (11,2) 16: (14,2) 18: (18)
%e 8: (6,2) 13: (10,3) 16: (13,3) 18: (15,3)
%e 8: (5,3) 13: (7,6) 16: (11,5) 18: (13,5)
%e 9: (7,2) 13: (6,5,2) 16: (11,3,2) 18: (13,3,2)
%e 9: (6,3) 14: (14) 16: (10,6) 18: (12,6)
%e 10: (10) 14: (12,2) 16: (7,6,3) 18: (11,7)
%e 10: (7,3) 14: (11,3) 16: (6,5,3,2) 18: (11,5,2)
%e 10: (5,3,2) 14: (7,5,2) 17: (17) 18: (10,6,2)
%e 11: (11) 14: (6,5,3) 17: (15,2) 18: (10,5,3)
%e 11: (6,5) 15: (15) 17: (14,3) 18: (7,6,5)
%t perpowQ[n_]:=GCD@@FactorInteger[n][[All,2]]>1;
%t Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&FreeQ[#,1]&&And@@Not/@perpowQ/@#&]],{n,20}]
%Y Cf. A001597, A007916, A025147, A052410, A087897, A305631, A321346, A323054, A323089, A323090.
%K nonn
%O 0,6
%A _Gus Wiseman_, Jan 04 2019
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