%I #24 Sep 17 2023 18:42:39

%S 1,1,2,3,4,6,8,12,14,20,25,34,40,54,64,85,98,127,149,189,219,277,316,

%T 395,456,557,638,778,889,1070,1226,1461,1667,1978,2250,2645,3019,3521,

%U 3997,4652,5267,6093,6909,7943,8982,10291,11609,13251,14947,16984,19104

%N Number of partitions of n such that no part is a sum of two other parts.

%C These are partitions containing the sum of no 2-element submultiset of the parts, a variation of binary sum-free partitions where parts cannot be re-used, ranked by A364461. The complement is counted by A237113. The non-binary version is A237667. For re-usable parts we have A364345. - _Gus Wiseman_, Aug 09 2023

%F a(n) = A000041(n) - A237113(n).

%e Of the 11 partitions of 6, only these 3 include a part that is a sum of two other parts: [3,2,1], [2,2,1,1], [2,1,1,1,1]. Thus, a(6) = 11 - 3 = 8.

%e From _Gus Wiseman_, Aug 09 2023: (Start)

%e The a(1) = 1 through a(8) = 14 partitions:

%e (1) (2) (3) (4) (5) (6) (7) (8)

%e (11) (21) (22) (32) (33) (43) (44)

%e (111) (31) (41) (42) (52) (53)

%e (1111) (221) (51) (61) (62)

%e (311) (222) (322) (71)

%e (11111) (411) (331) (332)

%e (3111) (421) (521)

%e (111111) (511) (611)

%e (2221) (2222)

%e (4111) (3311)

%e (31111) (5111)

%e (1111111) (41111)

%e (311111)

%e (11111111)

%e (End)

%t z = 20; t = Map[Count[Map[Length[Cases[Map[Total[#] &, Subsets[#, {2}]], Apply[Alternatives, #]]] &, IntegerPartitions[#]], 0] &, Range[z]] (* A236912 *)

%t u = PartitionsP[Range[z]] - t (* A237113, _Peter J. C. Moses_, Feb 03 2014 *)

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

%Y For subsets of {1..n} we have A085489, complement A088809.

%Y The complement is counted by A237113, ranks A364462.

%Y The non-binary version is A237667, ranks A364531.

%Y The non-binary complement is A237668, ranks A364532.

%Y The version with re-usable parts is A364345, ranks A364347.

%Y The (strict) version for linear combinations of parts is A364350.

%Y These partitions have ranks A364461.

%Y The strict case is A364533, non-binary A364349.

%Y The strict complement is A364670, with re-usable parts A363226.

%Y A000041 counts partitions, strict A000009.

%Y A008284 counts partitions by length, strict A008289.

%Y A108917 counts knapsack partitions, ranks A299702.

%Y A323092 counts double-free partitions, ranks A320340.

%Y Cf. A002865, A007865, A151897, A275972, A325862, A326083, A363225, A363260, A364346, A364755.

%K nonn

%O 0,3

%A _Clark Kimberling_, Feb 01 2014

%E a(0)=1 prepended by _Alois P. Heinz_, Sep 17 2023