login
A322968
Number of integer partitions of n with no ones whose parts are all powers of the same squarefree number.
1
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.
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}]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jan 01 2019
STATUS
approved