A325862 -id:A325862 - cp's OEIS frontend

A364272 Number of strict integer partitions of n containing the sum of some subset of the parts. A variation of sum-full strict partitions.

0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 3, 1, 4, 3, 8, 6, 11, 10, 17, 16, 26, 25, 39, 39, 54, 60, 82, 84, 116, 126, 160, 177, 222, 242, 302, 337, 402, 453, 542, 601, 722, 803, 936, 1057, 1234, 1373, 1601, 1793, 2056, 2312, 2658, 2950, 3395, 3789, 4281, 4814, 5452, 6048

Offset: 0

Gus Wiseman, Aug 01 2023

nonn

First differs from A316402 at a(16) = 11 due to (7,5,3,1).

			The a(6) = 1 through a(16) = 11 partitions (A=10):
  (321) . (431) . (532)  (5321) (642)  (5431) (743)  (6432)  (853)
                  (541)         (651)  (6421) (752)  (6531)  (862)
                  (4321)        (5421) (7321) (761)  (7431)  (871)
                                (6321)        (5432) (7521)  (6532)
                                              (6431) (9321)  (6541)
                                              (6521) (54321) (7432)
                                              (7421)         (7621)
                                              (8321)         (8431)
                                                             (8521)
                                                             (A321)
                                                             (64321)

The non-strict complement is A237667, ranks A364531.
The non-strict version is A237668, ranks A364532.
The complement in strict partitions is A364349, binary A364533.
The linear combination-free version is A364350.
For subsets of {1..n} we have A364534, complement A151897.
The binary version is A364670, allowing re-used parts A363226.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A108917 counts knapsack partitions, strict A275972, ranks A299702.
A236912 counts binary sum-free partitions, complement A237113.
A323092 counts double-free partitions, ranks A320340.
Cf. A007865, A025065, A085489, A093971, A111133, A240861, A320347, A325862, A363225, A364346.

Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&Intersection[#, Total/@Subsets[#,{2,Length[#]}]]!={}&]],{n,0,30}]

A237113 Number of partitions of n such that some part is a sum of two other parts.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 3, 3, 8, 10, 17, 22, 37, 47, 71, 91, 133, 170, 236, 301, 408, 515, 686, 860, 1119, 1401, 1798, 2232, 2829, 3495, 4378, 5381, 6682, 8165, 10060, 12238, 14958, 18116, 22018, 26533, 32071, 38490, 46265, 55318, 66193, 78843, 93949, 111503, 132326

Offset: 0

Clark Kimberling, Feb 04 2014

nonn

These are partitions containing the sum of some 2-element submultiset of the parts, a variation of binary sum-full partitions where parts cannot be re-used, ranked by A364462. The complement is counted by A236912. The non-binary version is A237668. For re-usable parts we have A363225. - Gus Wiseman, Aug 10 2023

			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) = 3.
From _Gus Wiseman_, Aug 09 2023: (Start)
The a(0) = 0 through a(9) = 10 partitions:
  .  .  .  .  (211)  (2111)  (321)    (3211)    (422)      (3321)
                             (2211)   (22111)   (431)      (4221)
                             (21111)  (211111)  (3221)     (4311)
                                                (4211)     (5211)
                                                (22211)    (32211)
                                                (32111)    (42111)
                                                (221111)   (222111)
                                                (2111111)  (321111)
                                                           (2211111)
                                                           (21111111)
(End)

The complement for subsets is A085489, with re-usable parts A007865.
For subsets of {1..n} we have A088809, with re-usable parts A093971.
The complement is counted by A236912, ranks A364461.
The non-binary complement is A237667, ranks A364531.
The non-binary version is A237668, ranks A364532.
With re-usable parts we have A363225, ranks A364348.
The complement with re-usable parts is A364345, ranks A364347.
These partitions have ranks A364462.
The strict case is A364670, with re-usable parts A363226.
A000041 counts integer 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, A151897, A237984, A325862, A326083, A363260.

z = 20; t = Map[Count[Map[Length[Cases[Map[Total[#] &, Subsets[#, {2}]],  Apply[Alternatives, #]]] &, IntegerPartitions[#]], 0] &, Range[z]] (* A236912 *)
u = PartitionsP[Range[z]] - t  (* A237113, Peter J. C. Moses, Feb 03 2014 *)
Table[Length[Select[IntegerPartitions[n],Intersection[#,Total/@Subsets[#,{2}]]!={}&]],{n,0,30}] (* Gus Wiseman, Aug 09 2023 *)

a(n) = A000041(n) - A236912(n).

a(0)=0 prepended by Alois P. Heinz, Sep 17 2023

A236912 Number of partitions of n such that no part is a sum of two other parts.

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 8, 12, 14, 20, 25, 34, 40, 54, 64, 85, 98, 127, 149, 189, 219, 277, 316, 395, 456, 557, 638, 778, 889, 1070, 1226, 1461, 1667, 1978, 2250, 2645, 3019, 3521, 3997, 4652, 5267, 6093, 6909, 7943, 8982, 10291, 11609, 13251, 14947, 16984, 19104

Offset: 0

Clark Kimberling, Feb 01 2014

nonn

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

			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.
From _Gus Wiseman_, Aug 09 2023: (Start)
The a(1) = 1 through a(8) = 14 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)
                                     (3111)    (421)      (521)
                                     (111111)  (511)      (611)
                                               (2221)     (2222)
                                               (4111)     (3311)
                                               (31111)    (5111)
                                               (1111111)  (41111)
                                                          (311111)
                                                          (11111111)
(End)

For subsets of {1..n} we have A085489, complement A088809.
The complement is counted by A237113, ranks A364462.
The non-binary version is A237667, ranks A364531.
The non-binary complement is A237668, ranks A364532.
The version with re-usable parts is A364345, ranks A364347.
The (strict) version for linear combinations of parts is A364350.
These partitions have ranks A364461.
The strict case is A364533, non-binary A364349.
The strict complement is A364670, with re-usable parts A363226.
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, A151897, A275972, A325862, A326083, A363225, A363260, A364346, A364755.

z = 20; t = Map[Count[Map[Length[Cases[Map[Total[#] &, Subsets[#, {2}]],  Apply[Alternatives, #]]] &, IntegerPartitions[#]], 0] &, Range[z]] (* A236912 *)
u = PartitionsP[Range[z]] - t  (* A237113, Peter J. C. Moses, Feb 03 2014 *)
Table[Length[Select[IntegerPartitions[n],Intersection[#,Total/@Subsets[#,{2}]]=={}&]],{n,0,15}] (* Gus Wiseman, Aug 09 2023 *)

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

a(0)=1 prepended by Alois P. Heinz, Sep 17 2023

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

Original entry on oeis.org

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

Clark Kimberling, Feb 11 2014

nonn

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)

			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)

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

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.

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 *)

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

A353864 Number of rucksack partitions of n: every consecutive constant subsequence has a different sum.

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 8, 11, 14, 19, 25, 33, 39, 51, 65, 82, 101, 126, 154, 191, 232, 284, 343, 416, 496, 600, 716, 855, 1018, 1209, 1430, 1691, 1991, 2345, 2747, 3224, 3762, 4393, 5116, 5946, 6897, 7998, 9257, 10696, 12336, 14213, 16343, 18781, 21538, 24687, 28253, 32291, 36876, 42057

Offset: 0

Gus Wiseman, May 23 2022

nonn

In a knapsack partition (A108917), every submultiset has a different sum, so these are run-knapsack partitions or rucksack partitions for short. Another variation of knapsack partitions is A325862.

			The a(0) = 1 through a(7) = 11 partitions:
  ()  (1)  (2)   (3)    (4)     (5)      (6)       (7)
           (11)  (21)   (22)    (32)     (33)      (43)
                 (111)  (31)    (41)     (42)      (52)
                        (1111)  (221)    (51)      (61)
                                (311)    (222)     (322)
                                (11111)  (321)     (331)
                                         (411)     (421)
                                         (111111)  (511)
                                                   (2221)
                                                   (4111)
                                                   (1111111)

Knapsack partitions are counted by A108917, ranked by A299702.
The strong case is A353838, counted by A353837, complement A353839.
The perfect case is A353865, ranked by A353867.
These partitions are ranked by A353866.
A000041 counts partitions, strict A000009.
A300273 ranks collapsible partitions, counted by A275870.
A304442 counts partitions with all equal run-sums, ranked by A353833.
A353832 represents the operation of taking run-sums of a partition.
A353836 counts partitions by number of distinct run-sums.
A353840-A353846 pertain to partition run-sum trajectory.
A353852 ranks compositions with all distinct run-sums, counted by A353850.
A353863 counts partitions whose weak run-sums cover an initial interval.
Cf. A008284, A018818, A225485, A325239, A325277, A325280, A325862, A353834.

msubs[s_]:=Join@@@Tuples[Table[Take[t,i],{t,Split[s]},{i,0,Length[t]}]];
Table[Length[Select[IntegerPartitions[n],UnsameQ@@Total/@Select[msubs[#],SameQ@@#&]&]],{n,0,30}]

a(50)-a(53) from Robert Price, Apr 03 2025

A237668 Number of partitions of n such that some part is a sum of two or more other parts.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 4, 4, 10, 13, 23, 27, 49, 60, 93, 115, 170, 210, 300, 370, 510, 632, 846, 1031, 1359, 1670, 2159, 2630, 3355, 4082, 5130, 6220, 7739, 9360, 11555, 13889, 16991, 20402, 24824, 29636, 35855, 42707, 51309, 60955, 72896, 86328, 102826, 121348

Offset: 0

Clark Kimberling, Feb 11 2014

nonn

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

			a(6) = 4 counts these partitions: 123, 1113, 1122, 11112.
From _Gus Wiseman_, Aug 12 2023: (Start)
The a(0) = 0 through a(9) = 13 partitions:
  .  .  .  .  (211)  (2111)  (321)    (3211)    (422)      (3321)
                             (2211)   (22111)   (431)      (4221)
                             (3111)   (31111)   (3221)     (4311)
                             (21111)  (211111)  (4211)     (5211)
                                                (22211)    (32211)
                                                (32111)    (33111)
                                                (41111)    (42111)
                                                (221111)   (222111)
                                                (311111)   (321111)
                                                (2111111)  (411111)
                                                           (2211111)
                                                           (3111111)
                                                           (21111111)
(End)

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

Cf. A179009.
The binary complement is A236912, ranks A364461.
The binary version is A237113, ranks A364462.
The complement is counted by A237667, ranks A364531.
The binary version with re-usable parts is A363225, ranks A364348.
The strict case is A364272.
The binary complement with re-usable parts is A364345, ranks A364347.
These partitions have ranks A364532.
For subsets instead of partitions we have A364534, complement A151897.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A108917 counts knapsack partitions, ranks A299702.
A299701 counts distinct subset-sums of prime indices.
A323092 counts double-free partitions, ranks A320340.
Cf. A002865, A088809, A237984, A325862, A326083, A363226, A364670.

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
(* Peter J. C. Moses, Feb 10 2014 *)
Table[Length[Select[IntegerPartitions[n],Intersection[#,Total/@Subsets[#,{2,Length[#]}]]!={}&]],{n,0,15}] (* Gus Wiseman, Aug 12 2023 *)

a(21)-a(47) from Giovanni Resta, Feb 22 2014

A364534 Number of subsets of {1..n} containing some element equal to the sum of two or more distinct other elements. A variation of sum-full subsets without re-used elements.

Original entry on oeis.org

0, 0, 0, 1, 3, 10, 27, 68, 156, 357, 775, 1667, 3505, 7303, 15019, 30759, 62489, 126619, 255542, 514721, 1034425, 2076924, 4164650, 8346306, 16715847, 33467324, 66982798, 134040148, 268179417, 536510608, 1073226084, 2146759579, 4293930436, 8588485846, 17177799658

Offset: 0

Gus Wiseman, Aug 02 2023

nonn

			The a(0) = 0 through a(5) = 10 subsets:
  .  .  .  {1,2,3}  {1,2,3}    {1,2,3}
                    {1,3,4}    {1,3,4}
                    {1,2,3,4}  {1,4,5}
                               {2,3,5}
                               {1,2,3,4}
                               {1,2,3,5}
                               {1,2,4,5}
                               {1,3,4,5}
                               {2,3,4,5}
                               {1,2,3,4,5}

Andrew Howroyd, Table of n, a(n) for n = 0..85

The binary version is A088809, complement A085489.
The complement is counted by A151897.
The complement for partitions is A237667, ranks A364531.
For partitions we have A237668, ranks A364532.
For strict partitions we have A364272, complement A364349.
A108917 counts knapsack partitions, strict A275972.
A236912 counts sum-free partitions w/o re-used parts, complement A237113.
Cf. A007865, A093971, A323092, A325862, A326083, A363225, A364345, A364346, A364348, A364350, A364533, A364670.

Table[Length[Select[Subsets[Range[n]],Intersection[#,Total/@Subsets[#,{2,Length[#]}]]!={}&]],{n,0,10}]

a(n) = 2^n - A151897(n). - Andrew Howroyd, Jan 27 2024

a(16)-a(25) from Chai Wah Wu, Nov 14 2023

a(26) onwards (using A151897) added by Andrew Howroyd, Jan 27 2024

A364345 Number of integer partitions of n without any three parts (a,b,c) (repeats allowed) satisfying a + b = c. A variation of sum-free partitions.

Original entry on oeis.org

1, 1, 2, 2, 4, 5, 7, 10, 13, 16, 21, 27, 34, 43, 54, 67, 83, 102, 122, 151, 182, 218, 258, 313, 366, 443, 513, 611, 713, 844, 975, 1149, 1325, 1554, 1780, 2079, 2381, 2761, 3145, 3647, 4134, 4767, 5408, 6200, 7014, 8035, 9048, 10320, 11639, 13207, 14836, 16850

Offset: 0

Gus Wiseman, Jul 20 2023

nonn

			The a(1) = 1 through a(8) = 13 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (32)     (33)      (43)       (44)
                    (31)    (41)     (51)      (52)       (53)
                    (1111)  (311)    (222)     (61)       (62)
                            (11111)  (411)     (322)      (71)
                                     (3111)    (331)      (332)
                                     (111111)  (511)      (611)
                                               (4111)     (2222)
                                               (31111)    (3311)
                                               (1111111)  (5111)
                                                          (41111)
                                                          (311111)
                                                          (11111111)

For subsets of {1..n} instead of partitions we have A007865 (sum-free sets), differences A288728.
Without re-using parts we have A236912, complement A237113.
Allowing the sum of any number of parts gives A237667 (cf. A108917).
The complement is counted by A363225, strict A363226, for subsets A093971.
The strict case is A364346.
These partitions have ranks A364347, complement A364348.
A000041 counts partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A323092 counts double-free partitions, ranks A320340.
Cf. A002865, A025065, A026905, A111133, A240861, A275972, A320347, A325862, A326083, A363260.

Table[Length[Select[IntegerPartitions[n],Select[Tuples[Union[#],3],#[[1]]+#[[2]]==#[[3]]&]=={}&]],{n,0,30}]

A364349 Number of strict integer partitions of n containing the sum of no subset of the parts.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 5, 5, 8, 7, 11, 11, 15, 14, 21, 21, 28, 29, 38, 38, 51, 50, 65, 68, 82, 83, 108, 106, 130, 136, 163, 168, 206, 210, 248, 266, 307, 322, 381, 391, 457, 490, 553, 582, 675, 703, 797, 854, 952, 1000, 1147, 1187, 1331, 1437, 1564, 1656, 1869

Offset: 0

Gus Wiseman, Jul 29 2023

nonn

First differs from A275972 in counting (7,5,3,1), which is not knapsack.

			The partition y = (7,5,3,1) has no subset with sum in y, so is counted under a(16).
The partition y = (15,8,4,2,1) has subset {1,2,4,8} with sum in y, so is not counted under a(31).
The a(1) = 1 through a(9) = 8 partitions:
  (1)  (2)  (3)    (4)    (5)    (6)    (7)      (8)      (9)
            (2,1)  (3,1)  (3,2)  (4,2)  (4,3)    (5,3)    (5,4)
                          (4,1)  (5,1)  (5,2)    (6,2)    (6,3)
                                        (6,1)    (7,1)    (7,2)
                                        (4,2,1)  (5,2,1)  (8,1)
                                                          (4,3,2)
                                                          (5,3,1)
                                                          (6,2,1)

For subsets of {1..n} we have A151897, complement A364534.
The non-strict version is A237667, ranked by A364531.
The complement in strict partitions is counted by A364272.
The linear combination-free version is A364350.
The binary version is A364533, allowing re-used parts A364346.
A000041 counts partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A108917 counts knapsack partitions, strict A275972.
A236912 counts sum-free partitions (not re-using parts), complement A237113.
A323092 counts double-free partitions, ranks A320340.
Cf. A007865, A025065, A085489, A093971, A111133, A240861, A320347, A325862, A363226, A364345.

Table[Length[Select[IntegerPartitions[n],Function[ptn,UnsameQ@@ptn&&Select[Subsets[ptn,{2,Length[ptn]}],MemberQ[ptn,Total[#]]&]=={}]]],{n,0,30}]

A196723 Number of subsets of {1..n} (including empty set) such that the pairwise sums of distinct elements are all distinct.

Original entry on oeis.org

1, 2, 4, 8, 15, 28, 50, 86, 143, 236, 376, 594, 913, 1380, 2048, 3016, 4367, 6302, 8974, 12670, 17685, 24580, 33738, 46072, 62367, 83990, 112342, 149734, 198153, 261562, 343210, 448694, 583445, 756846, 976086, 1255658, 1607831, 2053186, 2610560, 3312040, 4183689

Offset: 0

Alois P. Heinz, Oct 06 2011

nonn

The number of subsets of {1..n} such that every orderless pair of (not necessarily distinct) elements has a different sum is A143823(n).

			a(4) = 15: {}, {1}, {2}, {3}, {4}, {1,2}, {1,3}, {1,4}, {2,3}, {2,4}, {3,4}, {1,2,3}, {1,2,4}, {1,3,4}, {2,3,4}.

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..110

Cf. A143823, A196719, A196720, A196721, A196722, A196724.
The subset case is A196723 (this sequence).
The maximal case is A325878.
The integer partition case is A325857.
The strict integer partition case is A325877.
Heinz numbers of the counterexamples are given by A325991.
Cf. A108917, A325858, A325862, A325863, A325864.

b:= proc(n, s) local sn, m;
      m:= nops(s);
      sn:= [s[], n];
      `if`(n<1, 1, b(n-1, s) +`if`(m*(m+1)/2 = nops(({seq(seq(
       sn[i]+sn[j], j=i+1..m+1), i=1..m)})), b(n-1, sn), 0))
    end:
a:= proc(n) option remember;
      b(n-1, [n]) +`if`(n=0, 0, a(n-1))
    end:
seq(a(n), n=0..20);

b[n_, s_] := b[n, s] = Module[{sn, m}, m = Length[s]; sn = Append[s, n]; If[n<1, 1, b[n-1, s] + If[m*(m+1)/2 == Length[ Union[ Flatten[ Table[ sn[[i]] + sn[[j]], {i, 1, m}, {j, i+1, m+1}]]]], b[n-1, sn], 0]]];
a[n_] := a[n] = b[n-1, {n}] + If[n == 0, 0, a[n-1]]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jan 31 2017, translated from Maple *)
Table[Length[Select[Subsets[Range[n]],UnsameQ@@Plus@@@Subsets[#,{2}]&]],{n,0,10}] (* Gus Wiseman, Jun 03 2019 *)

Edited by Gus Wiseman, Jun 03 2019

cp's OEIS Frontend

A364272 Number of strict integer partitions of n containing the sum of some subset of the parts. A variation of sum-full strict partitions.

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A237113 Number of partitions of n such that some part is a sum of two other parts.

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A236912 Number of partitions of n such that no part is a sum of two other parts.

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

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A353864 Number of rucksack partitions of n: every consecutive constant subsequence has a different sum.

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A237668 Number of partitions of n such that some part is a sum of two or more other parts.

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A364534 Number of subsets of {1..n} containing some element equal to the sum of two or more distinct other elements. A variation of sum-full subsets without re-used elements.

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A364345 Number of integer partitions of n without any three parts (a,b,c) (repeats allowed) satisfying a + b = c. A variation of sum-free partitions.

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