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A326625 Number of strict integer partitions of n whose geometric mean is an integer.

%I A326625 #5 Jul 15 2019 01:46:16
%S A326625 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 A326625 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 A326625 9,7,11,12,7,7,12,9,13,11,10,13,7,14
%N A326625 Number of strict integer partitions of n whose geometric mean is an integer.
%H A326625 Wikipedia, <a href="https://en.wikipedia.org/wiki/Geometric_mean">Geometric mean</a>
%e A326625 The a(63) = 9 partitions:
%e A326625   (63)  (36,18,9)  (54,4,3,2)   (36,18,6,2,1)   (36,9,8,6,3,1)
%e A326625         (48,12,3)  (27,24,8,4)  (18,16,12,9,8)
%e A326625                    (32,18,9,4)
%e A326625 The initial terms count the following partitions:
%e A326625    1: (1)
%e A326625    2: (2)
%e A326625    3: (3)
%e A326625    4: (4)
%e A326625    5: (5)
%e A326625    5: (4,1)
%e A326625    6: (6)
%e A326625    7: (7)
%e A326625    7: (4,2,1)
%e A326625    8: (8)
%e A326625    9: (9)
%e A326625   10: (10)
%e A326625   10: (9,1)
%e A326625   10: (8,2)
%e A326625   11: (11)
%e A326625   12: (12)
%e A326625   13: (13)
%e A326625   13: (9,4)
%e A326625   13: (9,3,1)
%e A326625   14: (14)
%e A326625   14: (8,4,2)
%e A326625   15: (15)
%e A326625   15: (12,3)
%e A326625   16: (16)
%t A326625 Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&IntegerQ[GeometricMean[#]]&]],{n,0,30}]
%Y A326625 Partitions whose geometric mean is an integer are A067539.
%Y A326625 Strict partitions whose average is an integer are A102627.
%Y A326625 Cf. A078174, A078175, A326027, A326567/A326568, A326623, A326624.
%K A326625 nonn
%O A326625 0,6
%A A326625 _Gus Wiseman_, Jul 14 2019