A326642 Number of non-constant integer partitions of n whose mean and geometric mean are both integers.

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

Offset: 0

Author

Gus Wiseman, Jul 16 2019

Keywords

• nonn

Comments

The Heinz numbers of these partitions are given by A326646.

Examples

			The a(30) = 7 partitions:
  (27,3)
  (24,6)
  (24,3,3)
  (16,8,2,2,2)
  (9,9,9,1,1,1)
  (8,8,8,2,2,2)
  (8,8,4,4,1,1,1,1,1,1)
		

Links

• Wikipedia, Geometric mean

Crossrefs

Partitions with integer mean and geometric mean are A326641.

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

Non-constant partitions with integer geometric mean are A326624.

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

Programs

Mathematica

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

Formula

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