A326625 Number of strict integer partitions of n whose geometric mean is an integer.

0, 1, 1, 1, 1, 2, 1, 2, 1, 1, 3, 1, 1, 3, 2, 2, 1, 2, 1, 2, 4, 3, 1, 2, 1, 4, 5, 2, 3, 3, 3, 5, 1, 3, 5, 5, 3, 4, 4, 7, 7, 5, 5, 2, 4, 2, 5, 7, 4, 6, 9, 5, 7, 7, 8, 7, 5, 11, 5, 9, 9, 9, 7, 9, 5, 13, 7, 9, 7, 11, 12, 7, 7, 12, 9, 13, 11, 10, 13, 7, 14

Offset

0,6

Links

Table of n, a(n) for n=0..80.

Wikipedia, Geometric mean

Example

The a(63) = 9 partitions:
  (63)  (36,18,9)  (54,4,3,2)   (36,18,6,2,1)   (36,9,8,6,3,1)
        (48,12,3)  (27,24,8,4)  (18,16,12,9,8)
                   (32,18,9,4)
The initial terms count the following partitions:
   1: (1)
   2: (2)
   3: (3)
   4: (4)
   5: (5)
   5: (4,1)
   6: (6)
   7: (7)
   7: (4,2,1)
   8: (8)
   9: (9)
  10: (10)
  10: (9,1)
  10: (8,2)
  11: (11)
  12: (12)
  13: (13)
  13: (9,4)
  13: (9,3,1)
  14: (14)
  14: (8,4,2)
  15: (15)
  15: (12,3)
  16: (16)

Mathematica

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

Crossrefs

Partitions whose geometric mean is an integer are A067539.

Strict partitions whose average is an integer are A102627.

Cf. A078174, A078175, A326027, A326567/A326568, A326623, A326624.

Sequence in context: A368210 A233932 A008289 * A188884 A116679 A350032

Adjacent sequences: A326622 A326623 A326624 * A326626 A326627 A326628

Keyword

nonn

Author

Gus Wiseman, Jul 14 2019

Status

approved