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

Author

Gus Wiseman, Apr 01 2019

Keywords

• nonn

Comments

The Heinz numbers of these partitions are given by A054744.

Examples

			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)
		

Crossrefs

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

Programs

Mathematica

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

Formula

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