A322968 Number of integer partitions of n with no ones whose parts are all powers of the same squarefree number.

1, 0, 1, 1, 2, 1, 4, 1, 4, 2, 6, 1, 9, 1, 8, 4, 10, 1, 14, 1, 16, 5, 16, 1, 24, 2, 22, 5, 28, 1, 37, 1, 36, 7, 38, 4, 55, 1, 48, 9, 63, 1, 73, 1, 76, 12, 76, 1, 105, 2, 98, 11, 116, 1, 128, 5, 143, 14, 142, 1, 186, 1, 168, 18, 202, 5

Offset

0,5

Comments

First differs from A072721 at a(12) = 9, A072721(12) = 10.

Links

Table of n, a(n) for n=0..65.

Example

The a(2) = 1 through a(12) = 9 integer partitions (A = 10, B = 11):
  (2)  (3)  (4)   (5)  (6)    (7)  (8)     (9)    (A)      (B)  (66)
            (22)       (33)        (44)    (333)  (55)          (84)
                       (42)        (422)          (82)          (93)
                       (222)       (2222)         (442)         (444)
                                                  (4222)        (822)
                                                  (22222)       (3333)
                                                                (4422)
                                                                (42222)
                                                                (222222)
The a(20) = 16 integer partitions:
  (10,10), (16,4),
  (8,8,4), (16,2,2),
  (5,5,5,5), (8,4,4,4), (8,8,2,2),
  (4,4,4,4,4), (8,4,4,2,2),
  (4,4,4,4,2,2), (8,4,2,2,2,2),
  (4,4,4,2,2,2,2), (8,2,2,2,2,2,2),
  (4,4,2,2,2,2,2,2),
  (4,2,2,2,2,2,2,2,2),
  (2,2,2,2,2,2,2,2,2,2).

Mathematica

radbase[n_]:=n^(1/GCD@@FactorInteger[n][[All, 2]]);
powsqfQ[n_]:=SameQ@@Last/@FactorInteger[n];
Table[Length[Select[IntegerPartitions[n], And[FreeQ[#, 1], And@@powsqfQ/@#, SameQ@@radbase/@#]&]], {n, 30}]

Crossrefs

Cf. A001597, A005117, A018819, A023893, A052410, A072720, A072721, A072774, A102430, A322900, A322903, A322911, A322912.

Sequence in context: A286518 A029205 A229340 * A072721 A339792 A285711

Adjacent sequences: A322965 A322966 A322967 * A322969 A322970 A322971

Keyword

nonn

Author

Gus Wiseman, Jan 01 2019

Status

approved