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%I A322527 #6 Dec 14 2018 21:20:35
%S A322527 1,1,2,3,5,7,11,13,18,21,31,34,45,51,63,72,88,97,120,128,158,174,201,
%T A322527 222,264,287,333,359,416,441,518,557,631,684,770,833,954,1017,1141,
%U A322527 1222,1378,1475,1643,1755,1939,2097,2327,2471,2758,2928,3233,3470,3813,4085
%N A322527 Number of integer partitions of n whose product of parts is a power of a squarefree number (A072774).
%e A322527 The a(1) = 1 through a(8) = 18 integer partitions:
%e A322527   (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
%e A322527        (11)  (21)   (22)    (32)     (33)      (52)       (44)
%e A322527              (111)  (31)    (41)     (42)      (61)       (53)
%e A322527                     (211)   (221)    (51)      (331)      (71)
%e A322527                     (1111)  (311)    (222)     (421)      (422)
%e A322527                             (2111)   (321)     (511)      (521)
%e A322527                             (11111)  (411)     (2221)     (611)
%e A322527                                      (2211)    (3211)     (2222)
%e A322527                                      (3111)    (4111)     (3311)
%e A322527                                      (21111)   (22111)    (4211)
%e A322527                                      (111111)  (31111)    (5111)
%e A322527                                                (211111)   (22211)
%e A322527                                                (1111111)  (32111)
%e A322527                                                           (41111)
%e A322527                                                           (221111)
%e A322527                                                           (311111)
%e A322527                                                           (2111111)
%e A322527                                                           (11111111)
%e A322527 Missing from the list for n = 7 through 9:
%e A322527   (43)   (62)    (54)
%e A322527   (322)  (332)   (63)
%e A322527          (431)   (432)
%e A322527          (3221)  (522)
%e A322527                  (621)
%e A322527                  (3222)
%e A322527                  (3321)
%e A322527                  (4311)
%e A322527                  (32211)
%t A322527 Table[Length[Select[IntegerPartitions[n],SameQ@@Last/@FactorInteger[Times@@#]&]],{n,30}]
%Y A322527 Cf. A003963, A005117, A038041, A062503, A064573, A072774, A295193, A302505, A306021, A319169, A320322, A322526, A322528, A322530.
%K A322527 nonn
%O A322527 0,3
%A A322527 _Gus Wiseman_, Dec 14 2018

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