A325132 Number of integer partitions of n where the multiplicity of each part k is at least prime(k).

1, 0, 1, 1, 1, 1, 2, 1, 3, 2, 4, 3, 5, 4, 6, 6, 7, 7, 10, 8, 11, 12, 12, 14, 17, 16, 20, 22, 24, 26, 31, 31, 37, 39, 43, 46, 54, 53, 63, 65, 73, 75, 87, 87, 100, 102, 115, 117, 133, 134, 151, 155, 172, 176, 197, 202, 223, 231, 254, 262, 290, 298, 327, 341, 370

Offset

0,7

Comments

The Heinz numbers of these partitions are given by A054744.

Links

Table of n, a(n) for n=0..64.

Formula

G.f.: Product_{k>=1} (1 + x^(prime(k)*k) / (1 - x^k)). - Ilya Gutkovskiy, Nov 28 2020

Example

The first few terms count the following integer partitions:
   0: ()
   2: (11)
   3: (111)
   4: (1111)
   5: (11111)
   6: (222)
   6: (111111)
   7: (1111111)
   8: (2222)
   8: (22211)
   8: (11111111)
   9: (222111)
   9: (111111111)
  10: (22222)
  10: (222211)
  10: (2221111)
  10: (1111111111)
  11: (2222111)
  11: (22211111)
  11: (11111111111)
  12: (222222)
  12: (2222211)
  12: (22221111)
  12: (222111111)
  12: (111111111111)

Mathematica

Table[Length[Select[IntegerPartitions[n], And@@Table[Count[#, i]>=Prime[i], {i, Union[#]}]&]], {n, 0, 30}]

Crossrefs

Cf. A052335, A054744, A062457, A087153, A117144, A324525, A324572.

Sequence in context: A030451 A241825 A029162 * A225854 A371973 A005044

Adjacent sequences: A325129 A325130 A325131 * A325133 A325134 A325135

Keyword

nonn

Author

Gus Wiseman, Apr 01 2019

Status

approved