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

0,3

Comments

The Heinz numbers of these partitions are given by A326645.

Links

Table of n, a(n) for n=0..75.

Wikipedia, Geometric mean

Example

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)

Mathematica

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

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.

Sequence in context: A169819 A134681 A218703 * A144372 A182861 A049238

Adjacent sequences: A326638 A326639 A326640 * A326642 A326643 A326644

Keyword

nonn

Author

Gus Wiseman, Jul 16 2019

Status

approved