OFFSET
1,7
COMMENTS
Also partitions such that (maximum) > 2*(mean).
These are partitions whose complement (see example) has size > n.
EXAMPLE
The a(7) = 3 through a(10) = 11 partitions:
(511) (611) (711) (721)
(4111) (5111) (5211) (811)
(31111) (41111) (6111) (6211)
(311111) (42111) (7111)
(51111) (52111)
(411111) (61111)
(3111111) (421111)
(511111)
(3211111)
(4111111)
(31111111)
The partition y = (3,2,1,1) has length 4 and maximum 3, and 4*3 is not > 2*7, so y is not counted under a(7).
The partition y = (4,2,1,1) has length 4 and maximum 4, and 4*4 is not > 2*8, so y is not counted under a(8).
The partition y = (5,1,1,1) has length 4 and maximum 5, and 4*5 > 2*8, so y is counted under a(8).
The partition y = (5,2,1,1) has length 4 and maximum 5, and 4*5 > 2*9, so y is counted under a(9).
The partition y = (3,2,1,1) has diagram:
o o o
o o .
o . .
o . .
with complement (shown in dots) of size 5, and 5 is not > 7, so y is not counted under a(7).
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], Length[#]*Max@@#>2n&]], {n, 30}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Mar 29 2023
STATUS
approved