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%I A322900 #9 Dec 31 2018 13:18:09
%S A322900 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 A322900 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 A322900 131,6,146,15,143,2,190,2,169,20,203,6,224,2,242,18,248
%N A322900 Number of integer partitions of n whose parts are all proper powers of the same number.
%C A322900 Such a partition contains either no 1's or only 1's.
%C A322900 A proper power of n is a number n^k for some positive integer k.
%C A322900 Also integer partitions whose parts all have the same radical base (A052410).
%e A322900 The a(1) = 1 through a(14) = 9 integer partitions (A = 10, B = 11, C = 12, D = 13, E = 14):
%e A322900   (1) (2)  (3)   (4)    (5)     (6)      (7)       (8)        (9)
%e A322900       (11) (111) (22)   (11111) (33)     (1111111) (44)       (333)
%e A322900                  (1111)         (42)               (422)      (111111111)
%e A322900                                 (222)              (2222)
%e A322900                                 (111111)           (11111111)
%e A322900 .
%e A322900   (A)          (B)           (C)            (D)             (E)
%e A322900   (55)         (11111111111) (66)           (1111111111111) (77)
%e A322900   (82)                       (84)                           (842)
%e A322900   (442)                      (93)                           (4442)
%e A322900   (4222)                     (444)                          (8222)
%e A322900   (22222)                    (822)                          (44222)
%e A322900   (1111111111)               (3333)                         (422222)
%e A322900                              (4422)                         (2222222)
%e A322900                              (42222)                        (11111111111111)
%e A322900                              (222222)
%e A322900                              (111111111111)
%t A322900 radbase[n_]:=n^(1/GCD@@FactorInteger[n][[All,2]]);
%t A322900 Table[Length[Select[IntegerPartitions[n],SameQ@@radbase/@#&]],{n,30}]
%Y A322900 a(n) = A072721(n) + 1.
%Y A322900 Cf. A000961, A001597, A018819, A023893, A023894, A052409, A052410, A072720, A102430, A302593, A322901, A322902, A322903.
%K A322900 nonn
%O A322900 0,3
%A A322900 _Gus Wiseman_, Dec 30 2018

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