A326642 - OEIS

%I #6 Jul 16 2019 22:01:56

%S 0,0,0,0,0,0,0,0,0,0,2,0,1,0,0,1,1,0,0,0,4,2,0,0,3,1,2,1,2,0,7,0,4,2,

%T 2,4,7,0,0,4,12,0,9,0,2,11,0,0,17,6,14,4,8,0,13,6,27,6,2,0,36,0,0,35,

%U 32,8,20,0,11,6,56,0,91,0,2,17

%N Number of non-constant integer partitions of n whose mean and geometric mean are both integers.

%C The Heinz numbers of these partitions are given by A326646.

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

%F a(n) = A326641(n) - A000005(n).

%e The a(30) = 7 partitions:

%e   (27,3)

%e   (24,6)

%e   (24,3,3)

%e   (16,8,2,2,2)

%e   (9,9,9,1,1,1)

%e   (8,8,8,2,2,2)

%e   (8,8,4,4,1,1,1,1,1,1)

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

%Y Partitions with integer mean and geometric mean are A326641.

%Y Heinz numbers of non-constant partitions with integer mean and geometric mean are A326646.

%Y Non-constant partitions with integer geometric mean are A326624.

%Y Cf. A051293, A067538, A067539, A102627, A316413, A326029, A326643, A326644, A326645, A326647.

%K nonn

%O 0,11

%A _Gus Wiseman_, Jul 16 2019