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Number of integer partitions of n that are empty, (1), or have at least two parts and these parts are pairwise coprime.
53

%I #21 Jan 18 2021 18:05:41

%S 1,1,1,2,3,5,6,9,11,14,17,22,26,32,37,42,50,59,69,80,91,101,115,133,

%T 152,170,190,210,235,265,300,334,366,398,441,484,541,597,648,703,770,

%U 848,935,1022,1102,1184,1281,1406,1534,1661,1789,1916,2062,2244,2435

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

%C The Heinz numbers of these partitions are given by A302696.

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

%H Fausto A. C. Cariboni, <a href="/A327516/b327516.txt">Table of n, a(n) for n = 0..750</a>

%H Gus Wiseman, <a href="https://docs.google.com/document/d/e/2PACX-1vSX9dPMGJhxB8rOknCGvOs6PiyhupdWNpqLsnphdgU6MEVqFBnWugAXidDhwHeKqZe_YnUqYeGOXsOk/pub">Sequences counting and encoding certain classes of multisets</a>

%F For n > 1, a(n) = A051424(n) - 1. - _Gus Wiseman_, Sep 18 2020

%e The a(1) = 1 through a(8) = 11 partitions:

%e (1) (11) (21) (31) (32) (51) (43) (53)

%e (111) (211) (41) (321) (52) (71)

%e (1111) (311) (411) (61) (431)

%e (2111) (3111) (511) (521)

%e (11111) (21111) (3211) (611)

%e (111111) (4111) (5111)

%e (31111) (32111)

%e (211111) (41111)

%e (1111111) (311111)

%e (2111111)

%e (11111111)

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

%Y Cf. A038348, A302569, A304709, A304711.

%Y A000837 is the relatively prime instead of pairwise coprime version.

%Y A051424 includes all singletons, with strict case A007360.

%Y A101268 is the ordered version (with singletons).

%Y A302696 ranks these partitions, with complement A335241.

%Y A305713 is the strict case.

%Y A307719 counts these partitions of length 3.

%Y A018783 counts partitions with a common divisor.

%Y A328673 counts pairwise non-coprime partitions.

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

%K nonn

%O 0,4

%A _Gus Wiseman_, Sep 19 2019