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

0,11

Comments

The Heinz numbers of these partitions are given by A326646.

Links

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

Wikipedia, Geometric mean

Formula

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

Example

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)

Mathematica

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

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.

Sequence in context: A296238 A309475 A221314 * A087371 A307597 A307056

Adjacent sequences: A326639 A326640 A326641 * A326643 A326644 A326645

Keyword

nonn

Author

Gus Wiseman, Jul 16 2019

Status

approved