cp's OEIS Frontend

A237667 Number of partitions of n such that no part is a sum of two or more other parts.

%I A237667 #24 Aug 11 2023 10:03:31
%S A237667 1,1,2,3,4,6,7,11,12,17,19,29,28,41,42,61,61,87,85,120,117,160,156,
%T A237667 224,216,288,277,380,363,483,474,622,610,783,755,994,986,1235,1191,
%U A237667 1549,1483,1876,1865,2306,2279,2806,2732,3406,3413,4091,4013,4991,4895,5872
%N A237667 Number of partitions of n such that no part is a sum of two or more other parts.
%C A237667 From _Gus Wiseman_, Aug 09 2023: (Start)
%C A237667 Includes all knapsack partitions (A108917), but first differs at a(12) = 28, A108917(12) = 25. The difference is accounted for by the non-knapsack partitions: (4332), (5331), (33222).
%C A237667 These are partitions not containing the sum of any non-singleton submultiset of the parts, a variation of non-binary sum-free partitions where parts cannot be re-used, ranked by A364531. The complement is counted by A237668. The binary version is A236912. For re-usable parts we have A364350.
%C A237667 (End)
%H A237667 Giovanni Resta, <a href="/A237667/b237667.txt">Table of n, a(n) for n = 0..100</a>
%H A237667 Giovanni Resta, <a href="/A237667/a237667.c.txt">C program for computing a(0)-a(100)</a>
%e A237667 For n = 6, the nonqualifiers are 123, 1113, 1122, 11112, leaving a(6) = 7.
%e A237667 From _Gus Wiseman_, Aug 09 2023: (Start)
%e A237667 The partition y = (5,3,1,1) has submultiset (3,1,1) with sum in y, so is not counted under a(10).
%e A237667 The partition y = (5,3,3,1) has no non-singleton submultiset with sum in y, so is counted under a(12).
%e A237667 The a(1) = 1 through a(8) = 12 partitions:
%e A237667   (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
%e A237667        (11)  (21)   (22)    (32)     (33)      (43)       (44)
%e A237667              (111)  (31)    (41)     (42)      (52)       (53)
%e A237667                     (1111)  (221)    (51)      (61)       (62)
%e A237667                             (311)    (222)     (322)      (71)
%e A237667                             (11111)  (411)     (331)      (332)
%e A237667                                      (111111)  (421)      (521)
%e A237667                                                (511)      (611)
%e A237667                                                (2221)     (2222)
%e A237667                                                (4111)     (3311)
%e A237667                                                (1111111)  (5111)
%e A237667                                                           (11111111)
%e A237667 (End)
%t A237667 Map[Count[Map[MemberQ[#,Apply[Alternatives,Map[Apply[Plus,#]&, DeleteDuplicates[DeleteCases[Subsets[#],_?(Length[#]<2&)]]]]]&, IntegerPartitions[#]],False]&,Range[20]]  (* _Peter J. C. Moses_, Feb 10 2014 *)
%t A237667 Table[Length[Select[IntegerPartitions[n],Intersection[#,Total/@Subsets[#,{2,Length[#]}]]=={}&]],{n,0,15}] (* _Gus Wiseman_, Aug 09 2023 *)
%Y A237667 For subsets of {1..n} we have A151897, binary A085489.
%Y A237667 The binary version is A236912, ranks A364461.
%Y A237667 The binary complement is A237113, ranks A364462.
%Y A237667 The complement is counted by A237668, ranks A364532.
%Y A237667 The binary version with re-usable parts is A364345, strict A364346.
%Y A237667 The strict case is A364349, binary A364533.
%Y A237667 These partitions have ranks A364531.
%Y A237667 The complement for subsets is A364534, binary A088809.
%Y A237667 A000041 counts partitions, strict A000009.
%Y A237667 A008284 counts partitions by length, strict A008289.
%Y A237667 A108917 counts knapsack partitions, ranks A299702.
%Y A237667 A323092 counts double-free partitions, ranks A320340.
%Y A237667 Cf. A002865, A007865, A179009, A325862, A326083, A363225, A363226, A364348, A364350, A364670.
%K A237667 nonn
%O A237667 0,3
%A A237667 _Clark Kimberling_, Feb 11 2014
%E A237667 a(21)-a(53) from _Giovanni Resta_, Feb 22 2014