A327516 Number of integer partitions of n that are empty, (1), or have at least two parts and these parts are pairwise coprime.

1, 1, 1, 2, 3, 5, 6, 9, 11, 14, 17, 22, 26, 32, 37, 42, 50, 59, 69, 80, 91, 101, 115, 133, 152, 170, 190, 210, 235, 265, 300, 334, 366, 398, 441, 484, 541, 597, 648, 703, 770, 848, 935, 1022, 1102, 1184, 1281, 1406, 1534, 1661, 1789, 1916, 2062, 2244, 2435

Offset: 0

Author

Gus Wiseman, Sep 19 2019

Keywords

• nonn

Comments

The Heinz numbers of these partitions are given by A302696.

Note that the definition excludes partitions with repeated parts other than 1 (cf. A038348, A304709).

Examples

			The a(1) = 1 through a(8) = 11 partitions:
  (1)  (11)  (21)   (31)    (32)     (51)      (43)       (53)
             (111)  (211)   (41)     (321)     (52)       (71)
                    (1111)  (311)    (411)     (61)       (431)
                            (2111)   (3111)    (511)      (521)
                            (11111)  (21111)   (3211)     (611)
                                     (111111)  (4111)     (5111)
                                               (31111)    (32111)
                                               (211111)   (41111)
                                               (1111111)  (311111)
                                                          (2111111)
                                                          (11111111)
		

Links

• Fausto A. C. Cariboni, Table of n, a(n) for n = 0..750
• Gus Wiseman, Sequences counting and encoding certain classes of multisets

Crossrefs

Cf. A038348, A302569, A304709, A304711.

A000837 is the relatively prime instead of pairwise coprime version.

A051424 includes all singletons, with strict case A007360.

A101268 is the ordered version (with singletons).

A302696 ranks these partitions, with complement A335241.

A305713 is the strict case.

A307719 counts these partitions of length 3.

A018783 counts partitions with a common divisor.

A328673 counts pairwise non-coprime partitions.

Cf. A087087, A220377, A326675, A333227, A333228, A335238.

Programs

Mathematica

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

Formula

For n > 1, a(n) = A051424(n) - 1. - Gus Wiseman, Sep 18 2020