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%I #9 Dec 31 2018 13:18:09
%S 1,1,2,2,3,2,5,2,5,3,7,2,11,2,9,5,11,2,16,2,18,6,17,2,27,3,23,6,30,2,
%T 38,2,37,8,39,5,58,2,49,10,66,2,74,2,78,14,77,2,109,3,100,12,118,2,
%U 131,6,146,15,143,2,190,2,169,20,203,6,224,2,242,18,248
%N Number of integer partitions of n whose parts are all proper powers of the same number.
%C Such a partition contains either no 1's or only 1's.
%C A proper power of n is a number n^k for some positive integer k.
%C Also integer partitions whose parts all have the same radical base (A052410).
%e The a(1) = 1 through a(14) = 9 integer partitions (A = 10, B = 11, C = 12, D = 13, E = 14):
%e (1) (2) (3) (4) (5) (6) (7) (8) (9)
%e (11) (111) (22) (11111) (33) (1111111) (44) (333)
%e (1111) (42) (422) (111111111)
%e (222) (2222)
%e (111111) (11111111)
%e .
%e (A) (B) (C) (D) (E)
%e (55) (11111111111) (66) (1111111111111) (77)
%e (82) (84) (842)
%e (442) (93) (4442)
%e (4222) (444) (8222)
%e (22222) (822) (44222)
%e (1111111111) (3333) (422222)
%e (4422) (2222222)
%e (42222) (11111111111111)
%e (222222)
%e (111111111111)
%t radbase[n_]:=n^(1/GCD@@FactorInteger[n][[All,2]]);
%t Table[Length[Select[IntegerPartitions[n],SameQ@@radbase/@#&]],{n,30}]
%Y a(n) = A072721(n) + 1.
%Y Cf. A000961, A001597, A018819, A023893, A023894, A052409, A052410, A072720, A102430, A302593, A322901, A322902, A322903.
%K nonn
%O 0,3
%A _Gus Wiseman_, Dec 30 2018