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%I #5 Jan 02 2019 03:48:21
%S 1,0,1,1,2,1,4,1,4,2,6,1,9,1,8,4,10,1,14,1,16,5,16,1,24,2,22,5,28,1,
%T 37,1,36,7,38,4,55,1,48,9,63,1,73,1,76,12,76,1,105,2,98,11,116,1,128,
%U 5,143,14,142,1,186,1,168,18,202,5
%N Number of integer partitions of n with no ones whose parts are all powers of the same squarefree number.
%C First differs from A072721 at a(12) = 9, A072721(12) = 10.
%e The a(2) = 1 through a(12) = 9 integer partitions (A = 10, B = 11):
%e (2) (3) (4) (5) (6) (7) (8) (9) (A) (B) (66)
%e (22) (33) (44) (333) (55) (84)
%e (42) (422) (82) (93)
%e (222) (2222) (442) (444)
%e (4222) (822)
%e (22222) (3333)
%e (4422)
%e (42222)
%e (222222)
%e The a(20) = 16 integer partitions:
%e (10,10), (16,4),
%e (8,8,4), (16,2,2),
%e (5,5,5,5), (8,4,4,4), (8,8,2,2),
%e (4,4,4,4,4), (8,4,4,2,2),
%e (4,4,4,4,2,2), (8,4,2,2,2,2),
%e (4,4,4,2,2,2,2), (8,2,2,2,2,2,2),
%e (4,4,2,2,2,2,2,2),
%e (4,2,2,2,2,2,2,2,2),
%e (2,2,2,2,2,2,2,2,2,2).
%t radbase[n_]:=n^(1/GCD@@FactorInteger[n][[All,2]]);
%t powsqfQ[n_]:=SameQ@@Last/@FactorInteger[n];
%t Table[Length[Select[IntegerPartitions[n],And[FreeQ[#,1],And@@powsqfQ/@#,SameQ@@radbase/@#]&]],{n,30}]
%Y Cf. A001597, A005117, A018819, A023893, A052410, A072720, A072721, A072774, A102430, A322900, A322903, A322911, A322912.
%K nonn
%O 0,5
%A _Gus Wiseman_, Jan 01 2019
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