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%I #26 Aug 12 2023 09:42:40
%S 0,0,0,0,1,1,4,4,10,13,23,27,49,60,93,115,170,210,300,370,510,632,846,
%T 1031,1359,1670,2159,2630,3355,4082,5130,6220,7739,9360,11555,13889,
%U 16991,20402,24824,29636,35855,42707,51309,60955,72896,86328,102826,121348
%N Number of partitions of n such that some part is a sum of two or more other parts.
%C These are partitions containing the sum of some non-singleton submultiset of the parts, a variation of non-binary sum-full partitions where parts cannot be re-used, ranked by A364532. The complement is counted by A237667. The binary version is A237113, or A363225 with re-usable parts. This sequence is weakly increasing. - _Gus Wiseman_, Aug 12 2023
%H Giovanni Resta, <a href="/A237668/b237668.txt">Table of n, a(n) for n = 0..100</a>
%H Giovanni Resta, <a href="/A237668/a237668.c.txt">C program for computing a(0)-a(100)</a>
%e a(6) = 4 counts these partitions: 123, 1113, 1122, 11112.
%e From _Gus Wiseman_, Aug 12 2023: (Start)
%e The a(0) = 0 through a(9) = 13 partitions:
%e . . . . (211) (2111) (321) (3211) (422) (3321)
%e (2211) (22111) (431) (4221)
%e (3111) (31111) (3221) (4311)
%e (21111) (211111) (4211) (5211)
%e (22211) (32211)
%e (32111) (33111)
%e (41111) (42111)
%e (221111) (222111)
%e (311111) (321111)
%e (2111111) (411111)
%e (2211111)
%e (3111111)
%e (21111111)
%e (End)
%t z = 20; m = Map[Count[Map[MemberQ[#, Apply[Alternatives, Map[Apply[Plus, #] &, DeleteDuplicates[DeleteCases[Subsets[#], _?(Length[#] < 2 &)]]]]] &, IntegerPartitions[#]], False] &, Range[z]]; PartitionsP[Range[z]] - m
%t (* _Peter J. C. Moses_, Feb 10 2014 *)
%t Table[Length[Select[IntegerPartitions[n],Intersection[#,Total/@Subsets[#,{2,Length[#]}]]!={}&]],{n,0,15}] (* _Gus Wiseman_, Aug 12 2023 *)
%Y Cf. A179009.
%Y The binary complement is A236912, ranks A364461.
%Y The binary version is A237113, ranks A364462.
%Y The complement is counted by A237667, ranks A364531.
%Y The binary version with re-usable parts is A363225, ranks A364348.
%Y The strict case is A364272.
%Y The binary complement with re-usable parts is A364345, ranks A364347.
%Y These partitions have ranks A364532.
%Y For subsets instead of partitions we have A364534, complement A151897.
%Y A000041 counts integer partitions, strict A000009.
%Y A008284 counts partitions by length, strict A008289.
%Y A108917 counts knapsack partitions, ranks A299702.
%Y A299701 counts distinct subset-sums of prime indices.
%Y A323092 counts double-free partitions, ranks A320340.
%Y Cf. A002865, A088809, A237984, A325862, A326083, A363226, A364670.
%K nonn
%O 0,7
%A _Clark Kimberling_, Feb 11 2014
%E a(21)-a(47) from _Giovanni Resta_, Feb 22 2014
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