A326641 Number of integer partitions of n whose mean and geometric mean are both integers.

0, 1, 2, 2, 3, 2, 4, 2, 4, 3, 6, 2, 7, 2, 4, 5, 6, 2, 6, 2, 10, 6, 4, 2, 11, 4, 6, 5, 8, 2, 15, 2, 10, 6, 6, 8, 16, 2, 4, 8, 20, 2, 17, 2, 8, 17, 4, 2, 27, 9, 20, 8, 14, 2, 21, 10, 35, 10, 6, 2, 48, 2, 4, 41, 39, 12, 28, 2, 17, 10, 64, 2, 103, 2, 6, 23

Offset: 0

Author

Gus Wiseman, Jul 16 2019

Keywords

• nonn

Comments

The Heinz numbers of these partitions are given by A326645.

Examples

			The a(4) = 3 through a(10) = 6 partitions (A = 10):
  (4)     (5)      (6)       (7)        (8)         (9)          (A)
  (22)    (11111)  (33)      (1111111)  (44)        (333)        (55)
  (1111)           (222)                (2222)      (111111111)  (82)
                   (111111)             (11111111)               (91)
                                                                 (22222)
                                                                 (1111111111)
		

Links

• Wikipedia, Geometric mean

Crossrefs

Partitions with integer mean are A067538.

Partitions with integer geometric mean are A067539.

Non-constant partitions with integer mean and geometric mean are A326642.

Subsets with integer mean and geometric mean are A326643.

Heinz numbers of partitions with integer mean and geometric mean are A326645.

Strict partitions with integer mean and geometric mean are A326029.

Cf. A051293, A078175, A082553, A102627, A316413, A326027, A326623, A326644, A326646, A326647.

Programs

Mathematica

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