A335240 Number of integer partitions of n that are not pairwise coprime, where a singleton is not coprime unless it is (1).

1, 0, 1, 1, 2, 2, 5, 6, 11, 16, 25, 34, 51, 69, 98, 134, 181, 238, 316, 410, 536, 691, 887, 1122, 1423, 1788, 2246, 2800, 3483, 4300, 5304, 6508, 7983, 9745, 11869, 14399, 17436, 21040, 25367, 30482, 36568, 43735, 52239, 62239, 74073, 87950, 104277, 123348

Offset: 0

Author

Gus Wiseman, May 30 2020

Keywords

• nonn

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.

Examples

			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)
		

Links

• Fausto A. C. Cariboni, Table of n, a(n) for n = 0..750

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.

Cf. A007360, A101268, A302569, A335235, A335236, A335237, A335238.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],!CoprimeQ@@#&]],{n,0,30}]