cp's OEIS Frontend

A366754 Number of non-knapsack integer partitions of n.

%I A366754 #8 Nov 08 2023 07:50:40
%S A366754 0,0,0,0,1,1,4,4,10,13,23,27,52,60,94,118,175,213,310,373,528,643,862,
%T A366754 1044,1403,1699,2199,2676,3426,4131,5256,6295,7884,9479,11722,14047,
%U A366754 17296,20623,25142,29942,36299,43081,51950,61439,73668,87040,103748,122149,145155,170487
%N A366754 Number of non-knapsack integer partitions of n.
%C A366754 A multiset is non-knapsack if there exist two different submultisets with the same sum.
%F A366754 a(n) = A000041(n) - A108917(n).
%e A366754 The a(4) = 1 through a(9) = 13 partitions:
%e A366754   (211)  (2111)  (321)    (3211)    (422)      (3321)
%e A366754                  (2211)   (22111)   (431)      (4221)
%e A366754                  (3111)   (31111)   (3221)     (4311)
%e A366754                  (21111)  (211111)  (4211)     (5211)
%e A366754                                     (22211)    (32211)
%e A366754                                     (32111)    (33111)
%e A366754                                     (41111)    (42111)
%e A366754                                     (221111)   (222111)
%e A366754                                     (311111)   (321111)
%e A366754                                     (2111111)  (411111)
%e A366754                                                (2211111)
%e A366754                                                (3111111)
%e A366754                                                (21111111)
%t A366754 Table[Length[Select[IntegerPartitions[n], !UnsameQ@@Total/@Union[Subsets[#]]&]], {n,0,15}]
%Y A366754 The complement is counted by A108917, strict A275972, ranks A299702.
%Y A366754 These partitions have ranks A299729.
%Y A366754 The strict case is A316402.
%Y A366754 The binary version is A366753, ranks A366740.
%Y A366754 A000041 counts integer partitions, strict A000009.
%Y A366754 A276024 counts positive subset-sums of partitions, strict A284640.
%Y A366754 A304792 counts subset-sum of partitions, strict A365925.
%Y A366754 A365543 counts partitions with subset-sum k, complement A046663.
%Y A366754 A365661 counts strict partitions with subset-sum k, complement A365663.
%Y A366754 A366738 counts semi-sums of partitions, strict A366741.
%Y A366754 Cf. A002033, A006827, A122768, A126796, A238628, A365923, A365924, A367095.
%K A366754 nonn
%O A366754 0,7
%A A366754 _Gus Wiseman_, Nov 08 2023