A325864 -id:A325864 - cp's OEIS frontend

A143823 Number of subsets {x(1),x(2),...,x(k)} of {1,2,...,n} such that all differences |x(i)-x(j)| are distinct.

1, 2, 4, 7, 13, 22, 36, 57, 91, 140, 216, 317, 463, 668, 962, 1359, 1919, 2666, 3694, 5035, 6845, 9188, 12366, 16417, 21787, 28708, 37722, 49083, 63921, 82640, 106722, 136675, 174895, 222558, 283108, 357727, 451575, 567536, 712856, 890405, 1112081, 1382416, 1717540

Offset: 0

John W. Layman, Sep 02 2008

When the set {x(1),x(2),...,x(k)} satisfies the property that all differences |x(i)-x(j)| are distinct (or alternately, all the sums are distinct), then it is called a Sidon set. So this sequence is basically the number of Sidon subsets of {1,2,...,n}. - Sayan Dutta, Feb 15 2024
See A143824 for sizes of the largest subsets of {1,2,...,n} with the desired property.
Also the number of subsets of {1..n} such that every orderless pair of (not necessarily distinct) elements has a different sum. - Gus Wiseman, Jun 07 2019

			{1,2,4} is a subset of {1,2,3,4}, with distinct differences 2-1=1, 4-1=3, 4-2=2 between pairs of elements, so {1,2,4} is counted as one of the 13 subsets of {1,2,3,4} with the desired property.  Only 2^4-13=3 subsets of {1,2,3,4} do not have this property: {1,2,3}, {2,3,4}, {1,2,3,4}.
From _Gus Wiseman_, May 17 2019: (Start)
The a(0) = 1 through a(5) = 22 subsets:
  {}  {}   {}     {}     {}       {}
      {1}  {1}    {1}    {1}      {1}
           {2}    {2}    {2}      {2}
           {1,2}  {3}    {3}      {3}
                  {1,2}  {4}      {4}
                  {1,3}  {1,2}    {5}
                  {2,3}  {1,3}    {1,2}
                         {1,4}    {1,3}
                         {2,3}    {1,4}
                         {2,4}    {1,5}
                         {3,4}    {2,3}
                         {1,2,4}  {2,4}
                         {1,3,4}  {2,5}
                                  {3,4}
                                  {3,5}
                                  {4,5}
                                  {1,2,4}
                                  {1,2,5}
                                  {1,3,4}
                                  {1,4,5}
                                  {2,3,5}
                                  {2,4,5}
(End)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..100 (terms 0..60 from Alois P. Heinz, 61..81 from Vaclav Kotesovec)
Thomas Bloom, Problem 861, Erdős Problems.
Terence Tao, Erdős problem database, see no. 861.
Wikipedia, Sidon sequence.

First differences are A308251.
Second differences are A169942.
Row sums of A381476.
The maximal case is A325879.
The integer partition case is A325858.
The strict integer partition case is A325876.
Heinz numbers of the counterexamples are given by A325992.
Cf. A143824, A054578, A169947.
Cf. A000079, A108917, A143824, A169942, A308251, A325676, A325677, A325679, A325683, A325860, A325864, A241688, A241689, A241690.

b:= proc(n, s) local sn, m;
      if n<1 then 1
    else sn:= [s[], n];
         m:= nops(sn);
         `if`(m*(m-1)/2 = nops(({seq(seq(sn[i]-sn[j],
           j=i+1..m), i=1..m-1)})), b(n-1, sn), 0) +b(n-1, s)
      fi
    end:
a:= proc(n) option remember;
       b(n-1, [n]) +`if`(n=0, 0, a(n-1))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Sep 14 2011

b[n_, s_] := Module[{ sn, m}, If[n<1, 1, sn = Append[s, n]; m = Length[sn]; If[m*(m-1)/2 == Length[Table[sn[[i]] - sn[[j]], {i, 1, m-1}, {j, i+1, m}] // Flatten // Union], b[n-1, sn], 0] + b[n-1, s]]]; a[n_] := a[n] = b[n - 1, {n}] + If[n == 0, 0, a[n-1]]; Table [a[n], {n, 0, 30}] (* Jean-François Alcover, Nov 08 2015, after Alois P. Heinz *)
Table[Length[Select[Subsets[Range[n]],UnsameQ@@Abs[Subtract@@@Subsets[#,{2}]]&]],{n,0,15}] (* Gus Wiseman, May 17 2019 *)

from itertools import combinations
def is_sidon_set(s):
    allsums = []
    for i in range(len(s)):
        for j in range(i, len(s)):
            allsums.append(s[i] + s[j])
    if len(allsums)==len(set(allsums)):
        return True
    return False
def a(n):
    sidon_count = 0
    for r in range(n + 1):
        subsets = combinations(range(1, n + 1), r)
        for subset in subsets:
            if is_sidon_set(subset):
                sidon_count += 1
    return sidon_count
print([a(n) for n in range(20)]) # Sayan Dutta, Feb 15 2024

from functools import cache
def b(n, s):
    if n < 1: return 1
    sn = s + [n]
    m = len(sn)
    return (b(n-1, sn) if m*(m-1)//2 == len(set(sn[i]-sn[j] for i in range(m-1) for j in range(i+1, m))) else 0) + b(n-1, s)
@cache
def a(n): return b(n-1, [n]) + (0 if n==0 else a(n-1))
print([a(n) for n in range(31)]) # Michael S. Branicky, Feb 15 2024 after Alois P. Heinz

a(n) = A169947(n-1) + n + 1 for n>=2. - Nathaniel Johnston, Nov 12 2010

a(n) = A054578(n) + 1 for n>0. - Alois P. Heinz, Jan 17 2013

a(21)-a(29) from Nathaniel Johnston, Nov 12 2010

Corrected a(21)-a(29) and more terms from Alois P. Heinz, Sep 14 2011

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

A325862 Number of integer partitions of n such that every set of distinct parts has a different sum.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 10, 14, 19, 26, 34, 46, 58, 77, 93, 122, 146, 188, 217, 282, 327, 410, 470, 596, 673, 848, 947, 1178, 1325, 1629, 1798, 2213, 2444, 2962, 3247, 3935, 4292, 5149, 5579, 6674, 7247, 8590, 9221, 10964, 11804, 13870, 14843, 17480, 18675, 21866

Offset: 0

Gus Wiseman, May 31 2019

nonn

A knapsack partition (A108917, A299702) is an integer partition such that every submultiset has a different sum. The one non-knapsack partition counted under a(4) is (2,1,1).

			The a(1) = 1 through a(7) = 14 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)
       (11)  (21)   (22)    (32)     (33)      (43)
             (111)  (31)    (41)     (42)      (52)
                    (211)   (221)    (51)      (61)
                    (1111)  (311)    (222)     (322)
                            (2111)   (411)     (331)
                            (11111)  (2211)    (421)
                                     (3111)    (511)
                                     (21111)   (2221)
                                     (111111)  (4111)
                                               (22111)
                                               (31111)
                                               (211111)
                                               (1111111)
The three non-knapsack partitions counted under a(6) are:
  (2,2,1,1)
  (3,1,1,1)
  (2,1,1,1,1)

Dominates A108917.

Cf. A002033, A034444, A196723, A275972, A276024, A299702, A325592, A325856, A325863, A325864, A325865, A325877.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@Plus@@@Subsets[Union[#]]&]],{n,0,20}]

A325877 Number of strict integer partitions of n such that every orderless pair of distinct parts has a different sum.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 9, 12, 14, 18, 19, 26, 28, 36, 37, 50, 52, 67, 68, 89, 94, 115, 121, 151, 160, 195, 200, 247, 265, 312, 329, 386, 418, 487, 519, 600, 640, 742, 792, 901, 978, 1088, 1185, 1331, 1453, 1605, 1729, 1925, 2101, 2311, 2524, 2741, 3000

Offset: 0

Gus Wiseman, Jun 02 2019

nonn

The non-strict case is A325857.

			The a(1) = 1 through a(10) = 9 partitions (A = 10):
  (1)  (2)  (3)   (4)   (5)   (6)    (7)    (8)    (9)    (A)
            (21)  (31)  (32)  (42)   (43)   (53)   (54)   (64)
                        (41)  (51)   (52)   (62)   (63)   (73)
                              (321)  (61)   (71)   (72)   (82)
                                     (421)  (431)  (81)   (91)
                                            (521)  (432)  (532)
                                                   (531)  (541)
                                                   (621)  (631)
                                                          (721)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..450

The subset case is A196723.
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. A002033, A108917, A143823, A275972.
Cf. A325854, A325855, A325858, A325862, A325864, A325876, A325879.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&UnsameQ@@Plus@@@Subsets[Union[#],{2}]&]],{n,0,30}]

A308546 Number of double-closed subsets of {1..n}.

Original entry on oeis.org

1, 2, 3, 6, 8, 16, 24, 48, 60, 120, 180, 360, 480, 960, 1440, 2880, 3456, 6912, 10368, 20736, 27648, 55296, 82944, 165888, 207360, 414720, 622080, 1244160, 1658880, 3317760, 4976640, 9953280, 11612160, 23224320, 34836480, 69672960, 92897280

Offset: 0

Gus Wiseman, Jun 06 2019

nonn

These are subsets containing twice any element whose double is <= n.
Also the number of subsets of {1..n} containing half of every element that is even. For example, the a(6) = 24 subsets are:
  {}  {1}  {1,2}  {1,2,3}  {1,2,3,4}  {1,2,3,4,5}  {1,2,3,4,5,6}
      {3}  {1,3}  {1,2,4}  {1,2,3,5}  {1,2,3,4,6}
      {5}  {1,5}  {1,2,5}  {1,2,3,6}  {1,2,3,5,6}
           {3,5}  {1,3,5}  {1,2,4,5}
           {3,6}  {1,3,6}  {1,3,5,6}
                  {3,5,6}

			The a(6) = 24 subsets:
  {}  {4}  {2,4}  {1,2,4}  {1,2,4,5}  {1,2,3,4,6}  {1,2,3,4,5,6}
      {5}  {3,6}  {2,4,5}  {1,2,4,6}  {1,2,4,5,6}
      {6}  {4,5}  {2,4,6}  {2,3,4,6}  {2,3,4,5,6}
           {4,6}  {3,4,6}  {2,4,5,6}
           {5,6}  {3,5,6}  {3,4,5,6}
                  {4,5,6}

Charlie Neder, Table of n, a(n) for n = 0..500

Cf. A007865, A050291, A103580, A120641, A320340, A323092, A325864, A326020, A326076, A326083, A326115.

Table[Length[Select[Subsets[Range[n]],SubsetQ[#,Select[2*#,#<=n&]]&]],{n,0,10}]

From Charlie Neder, Jun 10 2019: (Start)
a(n) = Product_{k < n/2} (2 + floor(log_2(n/(2k+1)))).
a(0) = 1, a(n) = a(n-1) * (1 + 1/A001511(n)). (End)

a(21)-a(36) from Charlie Neder, Jun 10 2019

A364461 Positive integers such that if prime(a)*prime(b) is a divisor, prime(a+b) is not.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 64, 65, 66, 67, 68, 69, 71, 73, 74, 75, 76

Offset: 1

Gus Wiseman, Jul 27 2023

nonn

Also Heinz numbers of a type of sum-free partitions not allowing re-used parts, counted by A236912.

			The prime indices of 198 are {1,2,2,5}, which is sum-free even though it is not knapsack (A299702, A299729), so 198 is in the sequence.

Subsets of this type are counted by A085489, with re-usable parts A007865.
Subsets not of this type are counted by A093971, w/ re-usable parts A088809.
Partitions of this type are counted by A236912.
Allowing parts to be re-used gives A364347, counted by A364345.
The complement allowing parts to be re-used is A364348, counted by A363225.
The non-binary version allowing re-used parts is counted by A364350.
The complement is A364462, counted by A237113.
The non-binary version is A364531, counted by A237667, complement A364532.
A001222 counts prime indices.
A108917 counts knapsack partitions, ranks A299702.
A112798 lists prime indices, sum A056239.
Cf. A151897, A288728, A320340, A325862, A325864, A326083, A363226, A364346.

prix[n_]:=If[n==1,{}, Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Intersection[prix[#], Total/@Subsets[prix[#],{2}]]=={}&]

A325878 Number of maximal subsets of {1..n} such that every orderless pair of distinct elements has a different sum.

Original entry on oeis.org

1, 1, 1, 1, 4, 5, 8, 22, 40, 56, 78, 124, 222, 390, 616, 892, 1220, 1620, 2182, 3042, 4392, 6364, 9054, 12608, 16980, 22244, 28482, 36208, 45864, 58692, 75804, 98440, 128694, 168250, 218558, 281210, 357594, 449402, 560034, 693332, 853546, 1050118, 1293458, 1596144, 1975394

Offset: 0

Gus Wiseman, Jun 02 2019

nonn

			The a(1) = 1 through a(6) = 8 subsets:
  {1}  {1,2}  {1,2,3}  {1,2,3}  {1,2,4}    {1,2,3,5}
                       {1,2,4}  {2,3,4}    {1,2,3,6}
                       {1,3,4}  {2,4,5}    {1,2,4,6}
                       {2,3,4}  {1,2,3,5}  {1,3,4,5}
                                {1,3,4,5}  {1,3,5,6}
                                           {1,4,5,6}
                                           {2,3,4,6}
                                           {2,4,5,6}

Andrew Howroyd, Table of n, a(n) for n = 0..60

The subset case is A196723.
The integer partition case is A325857.
The strict integer partition case is A325877.
Heinz numbers of the counterexamples are given by A325991.
Cf. A002033, A108917, A143823, A143824, A276024.
Cf. A325858, A325859, A325864, A325865, A325867, A325879, A325880, A382398.

fasmax[y_]:=Complement[y,Union@@(Most[Subsets[#]]&/@y)];
Table[Length[fasmax[Select[Subsets[Range[n]],UnsameQ@@Plus@@@Subsets[Union[#],{2}]&]]],{n,0,10}]

a(n)={
   my(ismaxl(b,w)=for(k=1, n, if(!bittest(b,k) && !bitand(w,b< n, ismaxl(b,w),
         my(s=self()(k+1, r, b, w));
         if(!bitand(w,b<Andrew Howroyd, Mar 23 2025

a(21) onwards from Andrew Howroyd, Mar 23 2025

A364532 Positive integers with a prime index equal to the sum of prime indices of some nonprime divisor. Heinz numbers of a variation of sum-full partitions.

Original entry on oeis.org

12, 24, 30, 36, 40, 48, 60, 63, 70, 72, 80, 84, 90, 96, 108, 112, 120, 126, 132, 140, 144, 150, 154, 156, 160, 165, 168, 180, 189, 192, 198, 200, 204, 210, 216, 220, 224, 228, 240, 252, 264, 270, 273, 276, 280, 286, 288, 300, 308, 312, 315, 320, 324, 325, 330

Offset: 1

Gus Wiseman, Aug 01 2023

nonn

First differs from A299729 (non-knapsack) in lacking 525: {2,3,3,4}.
First differs from A325777 in having 462: {1,2,4,5} and lacking 675:{2,2,2,3,3}.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
These are the Heinz numbers of partitions containing the sum of some non-singleton submultiset.

			The terms together with their prime indices begin:
  12: {1,1,2}
  24: {1,1,1,2}
  30: {1,2,3}
  36: {1,1,2,2}
  40: {1,1,1,3}
  48: {1,1,1,1,2}
  60: {1,1,2,3}
  63: {2,2,4}
  70: {1,3,4}
  72: {1,1,1,2,2}
  80: {1,1,1,1,3}
  84: {1,1,2,4}
  90: {1,2,2,3}
  96: {1,1,1,1,1,2}

Partitions not of this type are counted by A237667, strict A364349.
Partitions of this type are counted by A237668, strict A364272.
The binary complement is A364461, re-usable A364347 (counted by A364345).
The binary version is A364462, re-usable A364348 (counted by A363225).
The complement is A364531.
Subsets of this type are counted by A364534, complement A151897.
A000005 counts divisors, nonprime A033273, composite A055212.
A001222 counts prime indices.
A108917 counts knapsack partitions, strict A275972, for subsets A325864.
A112798 lists prime indices, sum A056239.
A299701 counts distinct subset-sums of prime indices.
A299702 ranks knapsack partitions, complement A299729.
Cf. A085489, A088809, A093971, A236912, A237113, A301900, A326083, A364350.

Select[Range[100],Intersection[prix[#],Total/@Subsets[prix[#],{2,Length[prix[#]]}]]!={}&]

A364466 Number of subsets of {1..n} where some element is a difference of two consecutive elements.

Original entry on oeis.org

0, 0, 1, 2, 6, 14, 34, 74, 164, 345, 734, 1523, 3161, 6488, 13302, 27104, 55150, 111823, 226443, 457586, 923721, 1862183, 3751130, 7549354, 15184291, 30521675, 61322711, 123151315, 247230601, 496158486, 995447739, 1996668494, 4004044396, 8027966324, 16092990132, 32255168125

Offset: 0

Gus Wiseman, Jul 31 2023

nonn

In other words, the elements are not disjoint from their own first differences.

			The a(0) = 0 through a(5) = 14 subsets:
  .  .  {1,2}  {1,2}    {1,2}      {1,2}
               {1,2,3}  {2,4}      {2,4}
                        {1,2,3}    {1,2,3}
                        {1,2,4}    {1,2,4}
                        {1,3,4}    {1,2,5}
                        {1,2,3,4}  {1,3,4}
                                   {1,4,5}
                                   {2,3,5}
                                   {2,4,5}
                                   {1,2,3,4}
                                   {1,2,3,5}
                                   {1,2,4,5}
                                   {1,3,4,5}
                                   {1,2,3,4,5}

For differences of all pairs we have A093971, complement A196723.
For partitions we have A363260, complement A364467.
The complement is counted by A364463.
For subset-sums instead of differences we have A364534, complement A325864.
For strict partitions we have A364536, complement A364464.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A050291 counts double-free subsets, complement A088808.
A108917 counts knapsack partitions, strict A275972.
A325325 counts partitions with all distinct differences, strict A320347.
Cf. A011782, A212986, A237113, A325877, A325878, A326083, A363225.

Table[Length[Select[Subsets[Range[n]],Intersection[#,Differences[#]]!={}&]],{n,0,10}]

from itertools import combinations
def A364466(n): return sum(1 for l in range(n+1) for c in combinations(range(1,n+1),l) if not set(c).isdisjoint({c[i+1]-c[i] for i in range(l-1)})) # Chai Wah Wu, Sep 26 2023

a(n) = 2^n - A364463(n). - Chai Wah Wu, Sep 26 2023

a(21)-a(32) from Chai Wah Wu, Sep 26 2023

a(33)-a(35) from Chai Wah Wu, Sep 27 2023

A326017 Triangle read by rows where T(n,k) is the number of knapsack partitions of n with maximum k.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 2, 1, 1, 0, 1, 1, 1, 2, 1, 1, 0, 1, 1, 2, 3, 2, 1, 1, 0, 1, 1, 2, 1, 3, 2, 1, 1, 0, 1, 1, 2, 2, 4, 3, 2, 1, 1, 0, 1, 1, 2, 3, 1, 4, 3, 2, 1, 1, 0, 1, 1, 3, 3, 4, 6, 4, 3, 2, 1, 1, 0, 1, 1, 1, 1, 3, 1, 6, 4

Offset: 0

Gus Wiseman, Jun 03 2019

An integer partition is knapsack if every distinct submultiset has a different sum.

			Triangle begins:
  1
  0  1
  0  1  1
  0  1  1  1
  0  1  1  1  1
  0  1  1  2  1  1
  0  1  1  1  2  1  1
  0  1  1  2  3  2  1  1
  0  1  1  2  1  3  2  1  1
  0  1  1  2  2  4  3  2  1  1
  0  1  1  2  3  1  4  3  2  1  1
  0  1  1  3  3  4  6  4  3  2  1  1
  0  1  1  1  1  3  1  6  4  3  2  1  1
  0  1  1  3  3  5  4  7  6  4  3  2  1  1
  0  1  1  2  3  5  4  1  7  6  4  3  2  1  1
  0  1  1  2  3  4  6  6 11  7  6  4  3  2  1  1
Row n = 9 counts the following partitions:
  (111111111)  (22221)  (333)   (432)  (54)     (63)    (72)   (81)  (9)
                        (3222)  (441)  (522)    (621)   (711)
                                       (531)    (6111)
                                       (51111)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..10010
Fausto A. C. Cariboni, Conjectures on columns of T(n,k), Jun 05 2021.

Row sums are A108917.
Column k = 3 is A326034.
Cf. A002033, A196723, A275972, A276024.
Cf. A325592, A325857, A325862, A325863, A325864, A325877, A326016, A326018.

ks[n_]:=Select[IntegerPartitions[n],UnsameQ@@Total/@Union[Subsets[#]]&];
Table[Length[Select[ks[n],Length[#]==k==0||Max@@#==k&]],{n,0,15},{k,0,n}]

cp's OEIS Frontend

A143823 Number of subsets {x(1),x(2),...,x(k)} of {1,2,...,n} such that all differences |x(i)-x(j)| are distinct.

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

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A325862 Number of integer partitions of n such that every set of distinct parts has a different sum.

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A325877 Number of strict integer partitions of n such that every orderless pair of distinct parts has a different sum.

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A308546 Number of double-closed subsets of {1..n}.

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A364461 Positive integers such that if prime(a)*prime(b) is a divisor, prime(a+b) is not.

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A325878 Number of maximal subsets of {1..n} such that every orderless pair of distinct elements has a different sum.

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A364532 Positive integers with a prime index equal to the sum of prime indices of some nonprime divisor. Heinz numbers of a variation of sum-full partitions.