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

1, 1, 2, 3, 4, 6, 7, 11, 12, 17, 19, 29, 28, 41, 42, 61, 61, 87, 85, 120, 117, 160, 156, 224, 216, 288, 277, 380, 363, 483, 474, 622, 610, 783, 755, 994, 986, 1235, 1191, 1549, 1483, 1876, 1865, 2306, 2279, 2806, 2732, 3406, 3413, 4091, 4013, 4991, 4895, 5872

Offset: 0

Author

Clark Kimberling, Feb 11 2014

Keywords

• nonn

Comments

From Gus Wiseman, Aug 09 2023: (Start)

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).

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.

(End)

Examples

			For n = 6, the nonqualifiers are 123, 1113, 1122, 11112, leaving a(6) = 7.
From _Gus Wiseman_, Aug 09 2023: (Start)
The partition y = (5,3,1,1) has submultiset (3,1,1) with sum in y, so is not counted under a(10).
The partition y = (5,3,3,1) has no non-singleton submultiset with sum in y, so is counted under a(12).
The a(1) = 1 through a(8) = 12 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (32)     (33)      (43)       (44)
             (111)  (31)    (41)     (42)      (52)       (53)
                    (1111)  (221)    (51)      (61)       (62)
                            (311)    (222)     (322)      (71)
                            (11111)  (411)     (331)      (332)
                                     (111111)  (421)      (521)
                                               (511)      (611)
                                               (2221)     (2222)
                                               (4111)     (3311)
                                               (1111111)  (5111)
                                                          (11111111)
(End)
		

Links

• Giovanni Resta, Table of n, a(n) for n = 0..100
• Giovanni Resta, C program for computing a(0)-a(100)

Crossrefs

For subsets of {1..n} we have A151897, binary A085489.

The binary version is A236912, ranks A364461.

The binary complement is A237113, ranks A364462.

The complement is counted by A237668, ranks A364532.

The binary version with re-usable parts is A364345, strict A364346.

The strict case is A364349, binary A364533.

These partitions have ranks A364531.

The complement for subsets is A364534, binary A088809.

A000041 counts partitions, strict A000009.

A008284 counts partitions by length, strict A008289.

A108917 counts knapsack partitions, ranks A299702.

A323092 counts double-free partitions, ranks A320340.

Cf. A002865, A007865, A179009, A325862, A326083, A363225, A363226, A364348, A364350, A364670.

Programs

Mathematica

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 *)
Table[Length[Select[IntegerPartitions[n],Intersection[#,Total/@Subsets[#,{2,Length[#]}]]=={}&]],{n,0,15}] (* Gus Wiseman, Aug 09 2023 *)

Extensions

a(21)-a(53) from Giovanni Resta, Feb 22 2014