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Number of integer partitions of n whose product of parts is a power of a squarefree number (A072774).
7

%I #6 Dec 14 2018 21:20:35

%S 1,1,2,3,5,7,11,13,18,21,31,34,45,51,63,72,88,97,120,128,158,174,201,

%T 222,264,287,333,359,416,441,518,557,631,684,770,833,954,1017,1141,

%U 1222,1378,1475,1643,1755,1939,2097,2327,2471,2758,2928,3233,3470,3813,4085

%N Number of integer partitions of n whose product of parts is a power of a squarefree number (A072774).

%e The a(1) = 1 through a(8) = 18 integer partitions:

%e (1) (2) (3) (4) (5) (6) (7) (8)

%e (11) (21) (22) (32) (33) (52) (44)

%e (111) (31) (41) (42) (61) (53)

%e (211) (221) (51) (331) (71)

%e (1111) (311) (222) (421) (422)

%e (2111) (321) (511) (521)

%e (11111) (411) (2221) (611)

%e (2211) (3211) (2222)

%e (3111) (4111) (3311)

%e (21111) (22111) (4211)

%e (111111) (31111) (5111)

%e (211111) (22211)

%e (1111111) (32111)

%e (41111)

%e (221111)

%e (311111)

%e (2111111)

%e (11111111)

%e Missing from the list for n = 7 through 9:

%e (43) (62) (54)

%e (322) (332) (63)

%e (431) (432)

%e (3221) (522)

%e (621)

%e (3222)

%e (3321)

%e (4311)

%e (32211)

%t Table[Length[Select[IntegerPartitions[n],SameQ@@Last/@FactorInteger[Times@@#]&]],{n,30}]

%Y Cf. A003963, A005117, A038041, A062503, A064573, A072774, A295193, A302505, A306021, A319169, A320322, A322526, A322528, A322530.

%K nonn

%O 0,3

%A _Gus Wiseman_, Dec 14 2018