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

%I A327516 #21 Jan 18 2021 18:05:41
%S A327516 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 A327516 152,170,190,210,235,265,300,334,366,398,441,484,541,597,648,703,770,
%U A327516 848,935,1022,1102,1184,1281,1406,1534,1661,1789,1916,2062,2244,2435
%N A327516 Number of integer partitions of n that are empty, (1), or have at least two parts and these parts are pairwise coprime.
%C A327516 The Heinz numbers of these partitions are given by A302696.
%C A327516 Note that the definition excludes partitions with repeated parts other than 1 (cf. A038348, A304709).
%H A327516 Fausto A. C. Cariboni, <a href="/A327516/b327516.txt">Table of n, a(n) for n = 0..750</a>
%H A327516 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 A327516 For n > 1, a(n) = A051424(n) - 1. - _Gus Wiseman_, Sep 18 2020
%e A327516 The a(1) = 1 through a(8) = 11 partitions:
%e A327516   (1)  (11)  (21)   (31)    (32)     (51)      (43)       (53)
%e A327516              (111)  (211)   (41)     (321)     (52)       (71)
%e A327516                     (1111)  (311)    (411)     (61)       (431)
%e A327516                             (2111)   (3111)    (511)      (521)
%e A327516                             (11111)  (21111)   (3211)     (611)
%e A327516                                      (111111)  (4111)     (5111)
%e A327516                                                (31111)    (32111)
%e A327516                                                (211111)   (41111)
%e A327516                                                (1111111)  (311111)
%e A327516                                                           (2111111)
%e A327516                                                           (11111111)
%t A327516 Table[Length[Select[IntegerPartitions[n],#=={}||CoprimeQ@@#&]],{n,0,30}]
%Y A327516 Cf. A038348, A302569, A304709, A304711.
%Y A327516 A000837 is the relatively prime instead of pairwise coprime version.
%Y A327516 A051424 includes all singletons, with strict case A007360.
%Y A327516 A101268 is the ordered version (with singletons).
%Y A327516 A302696 ranks these partitions, with complement A335241.
%Y A327516 A305713 is the strict case.
%Y A327516 A307719 counts these partitions of length 3.
%Y A327516 A018783 counts partitions with a common divisor.
%Y A327516 A328673 counts pairwise non-coprime partitions.
%Y A327516 Cf. A087087, A220377, A326675, A333227, A333228, A335238.
%K A327516 nonn
%O A327516 0,4
%A A327516 _Gus Wiseman_, Sep 19 2019