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A325132
Number of integer partitions of n where the multiplicity of each part k is at least prime(k).
2
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.
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}]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Apr 01 2019
STATUS
approved