A323088 Number of strict integer partitions of n using numbers that are not perfect powers.

1, 0, 1, 1, 0, 2, 1, 2, 2, 2, 3, 3, 4, 5, 5, 7, 7, 9, 11, 11, 15, 16, 18, 22, 24, 27, 32, 34, 41, 45, 51, 59, 64, 75, 82, 94, 105, 116, 132, 146, 163, 183, 202, 225, 251, 277, 309, 341, 378, 417, 463, 510, 564, 622, 685, 754, 830, 914, 1001, 1103, 1207, 1325

Offset

0,6

Links

Table of n, a(n) for n=0..61.

Formula

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

Example

A list of all strict integer partitions using numbers that are not perfect powers begins:
   2: (2)        11: (6,3,2)    15: (13,2)       17: (12,5)
   3: (3)        12: (12)       15: (12,3)       17: (12,3,2)
   5: (5)        12: (10,2)     15: (10,5)       17: (11,6)
   5: (3,2)      12: (7,5)      15: (10,3,2)     17: (10,7)
   6: (6)        12: (7,3,2)    15: (7,6,2)      17: (10,5,2)
   7: (7)        13: (13)       15: (7,5,3)      17: (7,5,3,2)
   7: (5,2)      13: (11,2)     16: (14,2)       18: (18)
   8: (6,2)      13: (10,3)     16: (13,3)       18: (15,3)
   8: (5,3)      13: (7,6)      16: (11,5)       18: (13,5)
   9: (7,2)      13: (6,5,2)    16: (11,3,2)     18: (13,3,2)
   9: (6,3)      14: (14)       16: (10,6)       18: (12,6)
  10: (10)       14: (12,2)     16: (7,6,3)      18: (11,7)
  10: (7,3)      14: (11,3)     16: (6,5,3,2)    18: (11,5,2)
  10: (5,3,2)    14: (7,5,2)    17: (17)         18: (10,6,2)
  11: (11)       14: (6,5,3)    17: (15,2)       18: (10,5,3)
  11: (6,5)      15: (15)       17: (14,3)       18: (7,6,5)

Mathematica

perpowQ[n_]:=GCD@@FactorInteger[n][[All, 2]]>1;
Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&FreeQ[#, 1]&&And@@Not/@perpowQ/@#&]], {n, 20}]

Crossrefs

Cf. A001597, A007916, A025147, A052410, A087897, A305631, A321346, A323054, A323089, A323090.

Sequence in context: A084840 A029278 A125950 * A052954 A123505 A320779

Adjacent sequences: A323085 A323086 A323087 * A323089 A323090 A323091

Keyword

nonn

Author

Gus Wiseman, Jan 04 2019

Status

approved