A323089 Number of strict integer partitions of n using 1 and numbers that are not perfect powers.

1, 1, 1, 2, 1, 2, 3, 3, 4, 4, 5, 6, 7, 9, 10, 12, 14, 16, 20, 22, 26, 31, 34, 40, 46, 51, 59, 66, 75, 86, 96, 110, 123, 139, 157, 176, 199, 221, 248, 278, 309, 346, 385, 427, 476, 528, 586, 650, 719, 795, 880, 973, 1074, 1186, 1307, 1439, 1584, 1744, 1915, 2104

Offset

0,4

Links

Table of n, a(n) for n=0..59.

Formula

O.g.f.: (1 + x) * Product_{n in A007916} (1 + x^n).

Example

A list of all strict integer partitions using 1 and numbers that are not perfect powers begins:
  1: (1)         8: (5,2,1)      12: (12)         14: (14)
  2: (2)         9: (7,2)        12: (11,1)       14: (13,1)
  3: (3)         9: (6,3)        12: (10,2)       14: (12,2)
  3: (2,1)       9: (6,2,1)      12: (7,5)        14: (11,3)
  4: (3,1)       9: (5,3,1)      12: (7,3,2)      14: (11,2,1)
  5: (5)        10: (10)         12: (6,5,1)      14: (10,3,1)
  5: (3,2)      10: (7,3)        12: (6,3,2,1)    14: (7,6,1)
  6: (6)        10: (7,2,1)      13: (13)         14: (7,5,2)
  6: (5,1)      10: (6,3,1)      13: (12,1)       14: (6,5,3)
  6: (3,2,1)    10: (5,3,2)      13: (11,2)       14: (6,5,2,1)
  7: (7)        11: (11)         13: (10,3)       15: (15)
  7: (6,1)      11: (10,1)       13: (10,2,1)     15: (14,1)
  7: (5,2)      11: (7,3,1)      13: (7,6)        15: (13,2)
  8: (7,1)      11: (6,5)        13: (7,5,1)      15: (12,3)
  8: (6,2)      11: (6,3,2)      13: (7,3,2,1)    15: (12,2,1)
  8: (5,3)      11: (5,3,2,1)    13: (6,5,2)      15: (11,3,1)

Mathematica

perpowQ[n_]:=GCD@@FactorInteger[n][[All, 2]]>1;
Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&And@@Not/@perpowQ/@#&]], {n, 65}]

Crossrefs

Cf. A001597, A007916, A052410, A087897, A276431, A303707, A305630, A323088, A323090.

Sequence in context: A358025 A087188 A102885 * A239511 A264052 A138585

Adjacent sequences: A323086 A323087 A323088 * A323090 A323091 A323092

Keyword

nonn

Author

Gus Wiseman, Jan 04 2019

Status

approved