A326625 - OEIS

%I #5 Jul 15 2019 01:46:16

%S 0,1,1,1,1,2,1,2,1,1,3,1,1,3,2,2,1,2,1,2,4,3,1,2,1,4,5,2,3,3,3,5,1,3,

%T 5,5,3,4,4,7,7,5,5,2,4,2,5,7,4,6,9,5,7,7,8,7,5,11,5,9,9,9,7,9,5,13,7,

%U 9,7,11,12,7,7,12,9,13,11,10,13,7,14

%N Number of strict integer partitions of n whose geometric mean is an integer.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Geometric_mean">Geometric mean</a>

%e The a(63) = 9 partitions:

%e   (63)  (36,18,9)  (54,4,3,2)   (36,18,6,2,1)   (36,9,8,6,3,1)

%e         (48,12,3)  (27,24,8,4)  (18,16,12,9,8)

%e                    (32,18,9,4)

%e The initial terms count the following partitions:

%e    1: (1)

%e    2: (2)

%e    3: (3)

%e    4: (4)

%e    5: (5)

%e    5: (4,1)

%e    6: (6)

%e    7: (7)

%e    7: (4,2,1)

%e    8: (8)

%e    9: (9)

%e   10: (10)

%e   10: (9,1)

%e   10: (8,2)

%e   11: (11)

%e   12: (12)

%e   13: (13)

%e   13: (9,4)

%e   13: (9,3,1)

%e   14: (14)

%e   14: (8,4,2)

%e   15: (15)

%e   15: (12,3)

%e   16: (16)

%t Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&IntegerQ[GeometricMean[#]]&]],{n,0,30}]

%Y Partitions whose geometric mean is an integer are A067539.

%Y Strict partitions whose average is an integer are A102627.

%Y Cf. A078174, A078175, A326027, A326567/A326568, A326623, A326624.

%K nonn

%O 0,6

%A _Gus Wiseman_, Jul 14 2019