A325880 -id:A325880 - cp's OEIS frontend

A325864 Number of subsets of {1..n} of which every subset has a different sum.

1, 2, 4, 7, 13, 22, 36, 56, 91, 135, 211, 307, 446, 625, 882, 1194, 1677, 2238, 3031, 4001, 5460, 6995, 9302, 11921, 15424, 19554, 25032, 31005, 39170, 48251, 59917, 73093, 90831, 109271, 134049, 160922, 196109, 234179, 284157, 335933, 408390, 482597, 575109

Offset: 0

Gus Wiseman, Jun 01 2019

nonn

			The a(0) = 1 through a(4) = 13 subsets:
  {}  {}   {}     {}     {}
      {1}  {1}    {1}    {1}
           {2}    {2}    {2}
           {1,2}  {3}    {3}
                  {1,2}  {4}
                  {1,3}  {1,2}
                  {2,3}  {1,3}
                         {1,4}
                         {2,3}
                         {2,4}
                         {3,4}
                         {1,2,4}
                         {2,3,4}

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..150 (terms 0..60 from Giovanni Resta)

Cf. A002033, A108917, A143823, A196723, A275972.

Cf. A325860, A325865, A325866, A325867, A325877, A325878, A325879, A325880.

Table[Length[Select[Subsets[Range[n]],UnsameQ@@Plus@@@Subsets[#]&]],{n,0,10}]

a(18)-a(42) from Alois P. Heinz, Jun 03 2019

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

A325859 Number of maximal subsets of {1..n} such that every orderless pair of distinct elements has a different product.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 4, 4, 11, 11, 28, 28, 60, 60, 140, 241, 299, 299, 572, 572, 971

Offset: 0

Gus Wiseman, May 31 2019

			The a(1) = 1 through a(9) = 11 subsets:
  {1}  {12}  {123}  {1234}  {12345}  {2356}   {23567}   {123457}  {235678}
                                     {12345}  {123457}  {123578}  {1234579}
                                     {12456}  {124567}  {124567}  {1235789}
                                     {13456}  {134567}  {125678}  {1245679}
                                                        {134567}  {1256789}
                                                        {134578}  {1345679}
                                                        {135678}  {1345789}
                                                        {145678}  {1356789}
                                                        {234578}  {1456789}
                                                        {235678}  {2345789}
                                                        {245678}  {2456789}

The subset case is A196724.
The maximal case is A325859.
The integer partition case is A325856.
The strict integer partition case is A325855.
Heinz numbers of the counterexamples are given by A325993.
Cf. A002033, A108917, A143823, A275972, A325858, A325860, A325861, A325869, A325878, A325879, A325880.

fasmax[y_]:=Complement[y,Union@@(Most[Subsets[#]]&/@y)];
Table[Length[fasmax[Select[Subsets[Range[n]],UnsameQ@@Times@@@Subsets[#,{2}]&]]],{n,0,15}]

A325865 Number of maximal subsets of {1..n} of which every subset has a different sum.

Original entry on oeis.org

1, 1, 1, 3, 3, 6, 14, 23, 27, 40, 64, 104, 180, 275, 399, 554, 679, 872, 1117, 1431, 1920, 2520, 3530, 4751, 6644, 8855, 12021, 15461, 19939, 25109, 31656, 38750, 46204, 55650, 65942, 78045, 91304, 106592, 124761, 145701, 172343, 201217, 238739, 280601, 339746, 400394

Offset: 0

Gus Wiseman, Jun 01 2019

nonn

			The a(1) = 1 through a(6) = 14 subsets:
  {1}  {1,2}  {1,2}  {1,3}    {1,2,4}  {1,2,4}
              {1,3}  {1,2,4}  {1,2,5}  {1,2,5}
              {2,3}  {2,3,4}  {1,3,5}  {1,2,6}
                              {2,3,4}  {1,3,5}
                              {2,4,5}  {1,3,6}
                              {3,4,5}  {1,4,6}
                                       {2,3,4}
                                       {2,3,6}
                                       {2,4,5}
                                       {2,5,6}
                                       {3,4,5}
                                       {3,4,6}
                                       {3,5,6}
                                       {4,5,6}

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

Cf. A002033, A108917, A143823, A196723, A275972.

Cf. A325860, A325864, A325866, A325867, A325877, A325878, A325879, A325880.

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

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

a(18) onwards from Andrew Howroyd, Mar 23 2025

A326016 Number of knapsack partitions of n such that no addition of one part up to the maximum is knapsack.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 1, 1, 0, 3, 0, 0, 0, 1, 0, 8, 0, 8, 4, 3, 0, 11, 5, 3, 2, 5, 0, 29, 2, 9, 8, 20, 2

Offset: 1

Gus Wiseman, Jun 03 2019

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

The Heinz numbers of these partitions are given by A326018.

			The initial terms count the following partitions:
  15: (5,4,3,3)
  21: (7,6,5,3)
  21: (7,5,3,3,3)
  24: (8,7,6,3)
  25: (7,5,5,4,4)
  27: (9,8,7,3)
  27: (9,7,6,5)
  27: (8,7,3,3,3,3)
  31: (10,8,6,6,1)
  33: (11,9,7,3,3)
  33: (11,8,5,5,4)
  33: (11,7,6,6,3)
  33: (11,7,3,3,3,3,3)
  33: (11,5,5,4,4,4)
  33: (10,9,8,3,3)
  33: (10,8,6,6,3)
  33: (10,8,3,3,3,3,3)

Cf. A002033, A108917, A275972, A276024.

Cf. A325863, A325864, A325877, A325878, A325880, A326015, A326017, A326018.

sums[ptn_]:=sums[ptn]=If[Length[ptn]==1,ptn,Union@@(Join[sums[#],sums[#]+Total[ptn]-Total[#]]&/@Union[Table[Delete[ptn,i],{i,Length[ptn]}]])];
ksQ[y_]:=Length[sums[Sort[y]]]==Times@@(Length/@Split[Sort[y]]+1)-1;
maxks[n_]:=Select[IntegerPartitions[n],ksQ[#]&&Select[Table[Sort[Append[#,i]],{i,Range[Max@@#]}],ksQ]=={}&];
Table[Length[maxks[n]],{n,30}]

A325867 Number of maximal subsets of {1..n} containing n such that every subset has a different sum.

Original entry on oeis.org

1, 1, 2, 2, 4, 8, 10, 12, 17, 34, 45, 77, 99, 136, 166, 200, 238, 328, 402, 660, 674, 1166, 1331, 1966, 2335, 3286, 3527, 4762, 5383, 6900, 7543, 9087, 10149, 12239, 13569, 16452, 17867, 22869, 23977, 33881, 33820, 43423, 48090, 68683, 67347, 95176, 97917, 131666, 136205

Offset: 1

Gus Wiseman, Jun 01 2019

nonn

These are maximal strict knapsack partitions (A275972, A326015) organized by maximum rather than sum.

			The a(1) = 1 through a(8) = 12 subsets:
  {1}  {1,2}  {1,3}  {1,2,4}  {1,2,5}  {1,2,6}  {1,2,7}    {1,3,8}
              {2,3}  {2,3,4}  {1,3,5}  {1,3,6}  {1,3,7}    {1,5,8}
                              {2,4,5}  {1,4,6}  {1,4,7}    {5,7,8}
                              {3,4,5}  {2,3,6}  {1,5,7}    {1,2,4,8}
                                       {2,5,6}  {2,3,7}    {1,4,6,8}
                                       {3,4,6}  {2,4,7}    {2,3,4,8}
                                       {3,5,6}  {2,6,7}    {2,4,5,8}
                                       {4,5,6}  {4,5,7}    {2,4,7,8}
                                                {4,6,7}    {3,4,6,8}
                                                {3,5,6,7}  {3,6,7,8}
                                                           {4,5,6,8}
                                                           {4,6,7,8}

Fausto A. C. Cariboni, Table of n, a(n) for n = 1..150 (terms 1..121 from Bert Dobbelaere)

Cf. A002033, A108917, A143823, A143824, A196723, A275972.

Cf. A325860, A325861, A325864, A325865, A325866, A325867, A325880.

fasmax[y_]:=Complement[y,Union@@(Most[Subsets[#]]&)/@y];
Table[Length[fasmax[Select[Subsets[Range[n]],MemberQ[#,n]&&UnsameQ@@Plus@@@Subsets[#]&]]],{n,15}]

def f(p0, n, m, cm):
    full, t, p = True, 0, p0
    while p>k)&1)==0 and ((m<Bert Dobbelaere, Mar 07 2021

More terms from Bert Dobbelaere, Mar 07 2021

A326015 Number of strict knapsack partitions of n such that no superset with the same maximum is knapsack.

Original entry on oeis.org

1, 0, 1, 1, 1, 0, 1, 1, 3, 2, 4, 4, 5, 3, 3, 4, 6, 2, 7, 6, 13, 9, 19, 16, 27, 21, 40, 33, 47, 37, 54, 48, 66, 51, 65, 65, 77, 64, 80, 71, 96, 60, 106, 95, 112, 93, 152, 114, 191, 131, 242, 192, 303, 210, 366, 300, 482, 352, 581, 450, 713, 539, 882, 689, 995

Offset: 1

Gus Wiseman, Jun 03 2019

nonn

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

These are the subsets counted by A325867, ordered by sum rather than maximum.

			The a(1) = 1 through a(17) = 6 strict knapsack partitions (empty columns not shown):
  {1}  {2,1}  {3,1}  {3,2}  {4,2,1}  {5,2,1}  {4,3,2}  {6,3,1}  {5,4,2}
                                              {5,3,1}  {7,2,1}  {6,3,2}
                                              {6,2,1}           {6,4,1}
                                                                {7,3,1}
.
  {5,4,3}  {6,4,3}  {6,5,3}  {6,5,4}    {7,5,4}    {7,6,4}
  {7,3,2}  {6,5,2}  {8,5,1}  {7,6,2}    {9,4,3}    {9,5,3}
  {7,4,1}  {7,4,2}  {9,3,2}  {8,4,2,1}  {9,6,1}    {9,6,2}
  {8,3,1}  {7,5,1}                      {9,4,2,1}  {8,4,3,2}
           {9,3,1}                                 {9,5,2,1}
                                                   {10,4,2,1}

Fausto A. C. Cariboni, Table of n, a(n) for n = 1..600

Cf. A002033, A108917, A275972, A276024.

Cf. A325863, A325864, A325877, A325878, A325880, A326016, A326017, A326018.

ksQ[y_]:=UnsameQ@@Total/@Union[Subsets[y]]
maxsks[n_]:=Select[Select[IntegerPartitions[n],UnsameQ@@#&&ksQ[#]&],Select[Table[Append[#,i],{i,Complement[Range[Max@@#],#]}],ksQ]=={}&];
Table[Length[maxsks[n]],{n,30}]

A326018 Heinz numbers of knapsack partitions such that no addition of one part up to the maximum is knapsack.

Original entry on oeis.org

1925, 12155, 20995, 23375, 37145

Offset: 1

Gus Wiseman, Jun 03 2019

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
An integer partition is knapsack if every submultiset has a different sum.
The enumeration of these partitions by sum is given by A326016.

			The sequence of terms together with their prime indices begins:
   1925: {3,3,4,5}
  12155: {3,5,6,7}
  20995: {3,6,7,8}
  23375: {3,3,3,5,7}
  37145: {3,7,8,9}

Cf. A002033, A108917, A275972, A299702, A299729, A304793.

Cf. A325780, A325782, A325857, A325862, A325878, A325880, A326015, A326016.

ksQ[y_]:=UnsameQ@@Total/@Union[Subsets[y]];
Select[Range[2,200],With[{phm=If[#==1,{},Flatten[Cases[FactorInteger[#],{p_,k_}:>Table[PrimePi[p],{k}]]]]},ksQ[phm]&&Select[Table[Sort[Append[phm,i]],{i,Max@@phm}],ksQ]=={}]&]

A325866 Number of subsets of {1..n} containing n such that every subset has a different sum.

Original entry on oeis.org

1, 2, 3, 6, 9, 14, 20, 35, 44, 76, 96, 139, 179, 257, 312, 483, 561, 793, 970, 1459, 1535, 2307, 2619, 3503, 4130, 5478, 5973, 8165, 9081, 11666, 13176, 17738, 18440, 24778, 26873, 35187, 38070, 49978, 51776, 72457, 74207, 92512, 102210, 135571, 136786, 179604

Offset: 1

Gus Wiseman, Jun 01 2019

nonn

These are strict knapsack partitions (A275972) organized by maximum rather than sum.

			The a(1) = 1 through a(6) = 14 subsets:
  {1}  {2}    {3}    {4}      {5}      {6}
       {1,2}  {1,3}  {1,4}    {1,5}    {1,6}
              {2,3}  {2,4}    {2,5}    {2,6}
                     {3,4}    {3,5}    {3,6}
                     {1,2,4}  {4,5}    {4,6}
                     {2,3,4}  {1,2,5}  {5,6}
                              {1,3,5}  {1,2,6}
                              {2,4,5}  {1,3,6}
                              {3,4,5}  {1,4,6}
                                       {2,3,6}
                                       {2,5,6}
                                       {3,4,6}
                                       {3,5,6}
                                       {4,5,6}

Fausto A. C. Cariboni, Table of n, a(n) for n = 1..150

Cf. A108917, A143823, A143824, A196723, A275972.

Cf. A325860, A325864, A325865, A325867, A325877, A325878, A325880.

Table[Length[Select[Subsets[Range[n]],MemberQ[#,n]&&UnsameQ@@Plus@@@Subsets[#]&]],{n,10}]

a(18)-a(46) from Alois P. Heinz, Jun 03 2019

cp's OEIS Frontend

A325864 Number of subsets of {1..n} of which every subset has a different sum.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A325859 Number of maximal subsets of {1..n} such that every orderless pair of distinct elements has a different product.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

A325865 Number of maximal subsets of {1..n} of which every subset has a different sum.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A326016 Number of knapsack partitions of n such that no addition of one part up to the maximum is knapsack.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A325867 Number of maximal subsets of {1..n} containing n such that every subset has a different sum.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Python

Extensions

A326015 Number of strict knapsack partitions of n such that no superset with the same maximum is knapsack.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A326018 Heinz numbers of knapsack partitions such that no addition of one part up to the maximum is knapsack.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs