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 A325863 Number of integer partitions of n such that every distinct non-singleton submultiset has a different sum. 7
 1, 1, 2, 3, 5, 6, 9, 11, 15, 17, 24, 29, 31, 41, 51, 58, 67, 84, 91, 117, 117 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS A knapsack partition (A108917, A299702) is an integer partition such that every submultiset has a different sum. The one non-knapsack partition counted under a(4) is (2,1,1). LINKS EXAMPLE The partition (2,1,1,1) has non-singleton submultisets {1,2} and {1,1,1} with the same sum, so (2,1,1,1) is not counted under a(5). The a(1) = 1 through a(8) = 15 partitions:   (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)        (11)  (21)   (22)    (32)     (33)      (43)       (44)              (111)  (31)    (41)     (42)      (52)       (53)                     (211)   (221)    (51)      (61)       (62)                     (1111)  (311)    (222)     (322)      (71)                             (11111)  (321)     (331)      (332)                                      (411)     (421)      (422)                                      (3111)    (511)      (431)                                      (111111)  (2221)     (521)                                                (4111)     (611)                                                (1111111)  (2222)                                                           (3311)                                                           (5111)                                                           (41111)                                                           (11111111) The 10 non-knapsack partitions counted under a(12):   (7,6,1)   (7,5,2)   (7,4,3)   (7,5,1,1)   (7,4,2,1)   (7,3,3,1)   (7,3,2,2)   (7,4,1,1,1)   (7,2,2,2,1)   (7,1,1,1,1,1,1,1) MATHEMATICA Table[Length[Select[IntegerPartitions[n], UnsameQ@@Plus@@@Union[Subsets[#, {2, Length[#]}]]&]], {n, 0, 15}] CROSSREFS Dominates A108917. Cf. A002033, A055212, A143823, A196723, A276024, A299702, A325856, A325862, A325864, A325865, A325866, A325867, A325877. Sequence in context: A239010 A104738 A319469 * A028309 A242717 A026810 Adjacent sequences:  A325860 A325861 A325862 * A325864 A325865 A325866 KEYWORD nonn,more AUTHOR Gus Wiseman, May 31 2019 STATUS approved

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Last modified May 29 20:42 EDT 2020. Contains 334710 sequences. (Running on oeis4.)