OFFSET
0,5
COMMENTS
We use the Mathematica definition for CoprimeQ, so a singleton is not considered coprime unless it is (1).
These are also partitions that are a singleton or whose product is strictly greater than the LCM of their parts.
LINKS
Fausto A. C. Cariboni, Table of n, a(n) for n = 0..750
EXAMPLE
The a(2) = 1 through a(9) = 16 partitions:
(2) (3) (4) (5) (6) (7) (8) (9)
(22) (221) (33) (322) (44) (63)
(42) (331) (62) (333)
(222) (421) (332) (432)
(2211) (2221) (422) (441)
(22111) (2222) (522)
(3221) (621)
(3311) (3222)
(4211) (3321)
(22211) (4221)
(221111) (22221)
(32211)
(33111)
(42111)
(222111)
(2211111)
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], !CoprimeQ@@#&]], {n, 0, 30}]
CROSSREFS
The version for relatively prime instead of coprime is A018783.
The Heinz numbers of these partitions are the complement of A302696.
The complement is counted by A327516.
Singleton or pairwise coprime partitions are counted by A051424.
Singleton or pairwise coprime sets are ranked by A087087.
Numbers whose binary indices are pairwise coprime are A326675.
All of the following pertain to compositions in standard order (A066099):
- GCD is A326674.
- LCM is A333226.
- Coprime compositions are A333227.
- Compositions whose distinct parts are coprime are A333228.
- Non-coprime compositions are A335239.
KEYWORD
nonn
AUTHOR
Gus Wiseman, May 30 2020
STATUS
approved