cp's OEIS Frontend

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

%I A335240 #12 Feb 16 2021 10:51:32
%S A335240 1,0,1,1,2,2,5,6,11,16,25,34,51,69,98,134,181,238,316,410,536,691,887,
%T A335240 1122,1423,1788,2246,2800,3483,4300,5304,6508,7983,9745,11869,14399,
%U A335240 17436,21040,25367,30482,36568,43735,52239,62239,74073,87950,104277,123348
%N A335240 Number of integer partitions of n that are not pairwise coprime, where a singleton is not coprime unless it is (1).
%C A335240 We use the Mathematica definition for CoprimeQ, so a singleton is not considered coprime unless it is (1).
%C A335240 These are also partitions that are a singleton or whose product is strictly greater than the LCM of their parts.
%H A335240 Fausto A. C. Cariboni, <a href="/A335240/b335240.txt">Table of n, a(n) for n = 0..750</a>
%e A335240 The a(2) = 1 through a(9) = 16 partitions:
%e A335240   (2)  (3)  (4)   (5)    (6)     (7)      (8)       (9)
%e A335240             (22)  (221)  (33)    (322)    (44)      (63)
%e A335240                          (42)    (331)    (62)      (333)
%e A335240                          (222)   (421)    (332)     (432)
%e A335240                          (2211)  (2221)   (422)     (441)
%e A335240                                  (22111)  (2222)    (522)
%e A335240                                           (3221)    (621)
%e A335240                                           (3311)    (3222)
%e A335240                                           (4211)    (3321)
%e A335240                                           (22211)   (4221)
%e A335240                                           (221111)  (22221)
%e A335240                                                     (32211)
%e A335240                                                     (33111)
%e A335240                                                     (42111)
%e A335240                                                     (222111)
%e A335240                                                     (2211111)
%t A335240 Table[Length[Select[IntegerPartitions[n],!CoprimeQ@@#&]],{n,0,30}]
%Y A335240 The version for relatively prime instead of coprime is A018783.
%Y A335240 The Heinz numbers of these partitions are the complement of A302696.
%Y A335240 The complement is counted by A327516.
%Y A335240 Singleton or pairwise coprime partitions are counted by A051424.
%Y A335240 Singleton or pairwise coprime sets are ranked by A087087.
%Y A335240 Numbers whose binary indices are pairwise coprime are A326675.
%Y A335240 All of the following pertain to compositions in standard order (A066099):
%Y A335240 - GCD is A326674.
%Y A335240 - LCM is A333226.
%Y A335240 - Coprime compositions are A333227.
%Y A335240 - Compositions whose distinct parts are coprime are A333228.
%Y A335240 - Non-coprime compositions are A335239.
%Y A335240 Cf. A007360, A101268, A302569, A335235, A335236, A335237, A335238.
%K A335240 nonn
%O A335240 0,5
%A A335240 _Gus Wiseman_, May 30 2020