cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 14 results. Next

A196724 Number of subsets of {1..n} (including empty set) such that the pairwise products of distinct elements are all distinct.

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 58, 116, 212, 416, 720, 1440, 2340, 4680, 7920, 13024, 23328, 46656, 74168, 148336, 229856, 371424, 615304, 1230608, 1780224, 3401568, 5589360, 9468504, 14397744, 28795488, 40312128, 80624256, 131388480, 206363168, 335814288, 521401536
Offset: 0

Views

Author

Alois P. Heinz, Oct 06 2011

Keywords

Comments

The number of subsets of {1..n} such that every orderless pair of (not necessarily distinct) elements has a different product is A325860(n). - Gus Wiseman, Jun 03 2019

Examples

			a(6) = 58: from the 2^6=64 subsets of {1,2,3,4,5,6} only 6 do not have all the pairwise products of elements distinct: {1,2,3,6}, {2,3,4,6}, {1,2,3,4,6}, {1,2,3,5,6}, {2,3,4,5,6}, {1,2,3,4,5,6}.

Links

Crossrefs

Cf. A143823, A196719, A196720, A196721, A196722, A196723.
The subset case is A196724 (this sequence).
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. A292886, A293627, A325860, A325861.

Programs

  • Maple
    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);
  • Mathematica
    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@@Times@@@Subsets[#,{2}]&]],{n,0,10}] (* Gus Wiseman, Jun 03 2019 *)

Extensions

Name edited by Gus Wiseman, Jun 03 2019
a(33)-a(35) from Fausto A. C. Cariboni, Oct 05 2020

A325860 Number of subsets of {1..n} such that every pair of distinct elements has a different quotient.

Original entry on oeis.org

1, 2, 4, 8, 14, 28, 52, 104, 188, 308, 548, 1096, 1784, 3568, 6168, 10404, 16200, 32400, 49968, 99936, 155584, 256944, 433736, 867472, 1297504, 2026288, 3387216, 5692056, 8682912, 17365824, 25243200, 50486400, 78433056, 125191968, 206649216, 328195632
Offset: 0

Views

Author

Gus Wiseman, May 31 2019

Keywords

Comments

Also subsets of {1..n} such that every orderless pair of (not necessarily distinct) elements has a different product.

Examples

			The a(0) = 1 through a(4) = 14 subsets:
  {}  {}   {}    {}     {}
      {1}  {1}   {1}    {1}
           {2}   {2}    {2}
           {12}  {3}    {3}
                 {12}   {4}
                 {13}   {12}
                 {23}   {13}
                 {123}  {14}
                        {23}
                        {24}
                        {34}
                        {123}
                        {134}
                        {234}

Links

Crossrefs

The subset case is A325860.
The maximal case is A325861.
The integer partition case is A325853.
The strict integer partition case is A325854.
Heinz numbers of the counterexamples are given by A325994.
Cf. A002033, A108917, A143823, A196723, A196723, A196724, A325855, A325858, A325859, A325868, A325869.

Programs

  • Mathematica
    Table[Length[Select[Subsets[Range[n]],UnsameQ@@Divide@@@Subsets[#,{2}]&]],{n,0,20}]

Extensions

a(21)-a(25) from Alois P. Heinz, Jun 07 2019
a(26)-a(35) from Fausto A. C. Cariboni, Oct 04 2020

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

Views

Author

Gus Wiseman, Jun 02 2019

Keywords

Examples

			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}

Links

Crossrefs

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.

Programs

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

Extensions

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

A325879 Number of maximal subsets of {1..n} such that every ordered pair of distinct elements has a different difference.

Original entry on oeis.org

1, 1, 1, 3, 3, 6, 14, 20, 24, 36, 64, 110, 176, 238, 294, 370, 504, 736, 1086, 1592, 2240, 2982, 3788, 4700, 5814, 7322, 9396, 12336, 16552, 22192, 29310, 38046, 48368, 60078, 73722, 89416, 108208, 131310, 160624, 198002, 247408, 310410, 390924, 490818, 613344, 758518
Offset: 0

Views

Author

Gus Wiseman, Jun 02 2019

Keywords

Comments

Also the number of maximal subsets of {1..n} such that every orderless pair of (not necessarily distinct) elements has a different sum.

Examples

			The a(0) = 1 through a(7) = 20 subsets:
  {}  {1}  {1,2}  {1,2}  {2,3}    {1,2,4}  {1,2,4}  {1,2,4}
                  {1,3}  {1,2,4}  {1,2,5}  {1,2,5}  {1,2,6}
                  {2,3}  {1,3,4}  {1,3,4}  {1,2,6}  {1,3,4}
                                  {1,4,5}  {1,3,4}  {1,4,5}
                                  {2,3,5}  {1,3,6}  {1,4,6}
                                  {2,4,5}  {1,4,5}  {1,5,6}
                                           {1,4,6}  {2,3,5}
                                           {1,5,6}  {2,3,6}
                                           {2,3,5}  {2,3,7}
                                           {2,3,6}  {2,4,5}
                                           {2,4,5}  {2,4,7}
                                           {2,5,6}  {2,5,6}
                                           {3,4,6}  {2,6,7}
                                           {3,5,6}  {3,4,6}
                                                    {3,4,7}
                                                    {3,5,6}
                                                    {4,5,7}
                                                    {4,6,7}
                                                    {1,2,5,7}
                                                    {1,3,6,7}

Links

Crossrefs

The subset case is A143823.
The integer partition case is A325858.
The strict integer partition case is A325876.
Heinz numbers of the counterexamples are given by A325992.
Cf. A002033, A108917, A143824, A196723, A382397.
Cf. A325859, A325861, A325865, A325867, A325869, A325878, A325992.

Programs

  • Mathematica
    fasmax[y_]:=Complement[y,Union@@(Most[Subsets[#]]&/@y)];
Table[Length[fasmax[Select[Subsets[Range[n]],UnsameQ@@Subtract@@@Subsets[Union[#],{2}]&]]],{n,0,10}]
  • PARI
    a(n)={
  my(ismaxl(b,w)=for(k=1, n, if(!bittest(b,k) && !bitand(w,bitor(b,1< n, ismaxl(b,w),
         my(s=self()(k+1, b,w));
         b+=1<Andrew Howroyd, Mar 27 2025

Extensions

a(21)-a(45) from Fausto A. C. Cariboni, Feb 08 2022

A325861 Number of maximal subsets of {1..n} such that every pair of distinct elements has a different quotient.

Original entry on oeis.org

1, 1, 1, 1, 3, 3, 6, 6, 9, 13, 32, 32, 57, 57, 140, 229, 373, 373, 549, 549, 825
Offset: 0

Views

Author

Gus Wiseman, May 31 2019

Keywords

Examples

			The a(1) = 1 through a(9) = 13 subsets:
  {1}  {12}  {123}  {123}  {1235}  {1235}   {12357}   {23457}   {24567}
                    {134}  {1345}  {1256}   {12567}   {24567}   {123578}
                    {234}  {2345}  {2345}   {23457}   {123578}  {134567}
                                   {2356}   {23567}   {125678}  {134578}
                                   {2456}   {24567}   {134567}  {135678}
                                   {13456}  {134567}  {134578}  {145678}
                                                      {135678}  {145789}
                                                      {145678}  {234579}
                                                      {235678}  {235678}
                                                                {235789}
                                                                {345789}
                                                                {356789}
                                                                {1256789}

Crossrefs

The subset case is A325860.
The maximal case is A325861.
The integer partition case is A325853.
The strict integer partition case is A325854.
Heinz numbers of the counterexamples are given by A325994.
Cf. A002033, A103300, A143823, A196724, A325859, A325868, A325869.

Programs

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

A343652 Number of maximal pairwise coprime sets of divisors of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 2, 4, 1, 4, 1, 4, 2, 2, 1, 6, 2, 2, 3, 4, 1, 5, 1, 5, 2, 2, 2, 8, 1, 2, 2, 6, 1, 5, 1, 4, 4, 2, 1, 8, 2, 4, 2, 4, 1, 6, 2, 6, 2, 2, 1, 10, 1, 2, 4, 6, 2, 5, 1, 4, 2, 5, 1, 12, 1, 2, 4, 4, 2, 5, 1, 8, 4, 2, 1, 10, 2, 2
Offset: 1

Views

Author

Gus Wiseman, Apr 25 2021

Keywords

Comments

Also the number of maximal pairwise coprime sets of divisors > 1 of n. For example, the a(n) sets for n = 12, 30, 36, 60, 120 are:
{6} {30} {6} {30} {30}
{12} {2,15} {12} {60} {60}
{2,3} {3,10} {18} {2,15} {120}
{3,4} {5,6} {36} {3,10} {2,15}
{2,3,5} {2,3} {3,20} {3,10}
{2,9} {4,15} {3,20}
{3,4} {5,6} {3,40}
{4,9} {5,12} {4,15}
{2,3,5} {5,6}
{3,4,5} {5,12}
{5,24}
{8,15}
{2,3,5}
{3,4,5}
{3,5,8}

Examples

			The a(n) sets for n = 12, 30, 36, 60, 120:
  {1,6}    {1,30}     {1,6}    {1,30}     {1,30}
  {1,12}   {1,2,15}   {1,12}   {1,60}     {1,60}
  {1,2,3}  {1,3,10}   {1,18}   {1,2,15}   {1,120}
  {1,3,4}  {1,5,6}    {1,36}   {1,3,10}   {1,2,15}
           {1,2,3,5}  {1,2,3}  {1,3,20}   {1,3,10}
                      {1,2,9}  {1,4,15}   {1,3,20}
                      {1,3,4}  {1,5,6}    {1,3,40}
                      {1,4,9}  {1,5,12}   {1,4,15}
                               {1,2,3,5}  {1,5,6}
                               {1,3,4,5}  {1,5,12}
                                          {1,5,24}
                                          {1,8,15}
                                          {1,2,3,5}
                                          {1,3,4,5}
                                          {1,3,5,8}

Crossrefs

The case of pairs is A063647.
The case of triples is A066620.
The non-maximal version counting empty sets and singletons is A225520.
The non-maximal version with no 1's is A343653.
The non-maximal version is A343655.
The version for subsets of {1..n} is A343659.
The case without 1's or singletons is A343660.
A018892 counts pairwise coprime unordered pairs of divisors.
A048691 counts pairwise coprime ordered pairs of divisors.
A048785 counts pairwise coprime ordered triples of divisors.
A084422, A187106, A276187, and A320426 count pairwise coprime sets.
A100565 counts pairwise coprime unordered triples of divisors.
A305713 counts pairwise coprime non-singleton strict partitions.
A324837 counts minimal subsets of {1...n} with least common multiple n.
A325683 counts maximal Golomb rulers.
A326077 counts maximal pairwise indivisible sets.
Cf. A005361, A007359, A051026, A062319, A067824, A074206, A146291, A285572, A325859, A326359, A326496, A326675, A343654.

Programs

  • Mathematica
    fasmax[y_]:=Complement[y,Union@@Most@*Subsets/@y];
Table[Length[fasmax[Select[Subsets[Divisors[n]],CoprimeQ@@#&]]],{n,100}]

Formula

a(n) = A343660(n) + A005361(n).

A325856 Number of integer partitions of n such that every pair of distinct parts has a different product.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 76, 100, 133, 171, 225, 287, 369, 467, 592, 740, 931, 1155, 1435, 1767, 2178, 2661, 3254, 3953, 4798, 5793, 6991, 8390, 10069, 12022, 14346, 17054, 20255, 23960, 28334, 33390, 39308, 46148, 54116, 63295, 73967, 86224
Offset: 0

Views

Author

Gus Wiseman, May 31 2019

Keywords

Examples

			The five partitions of 15 not satisfying the condition are:
  (8,4,2,1)
  (6,4,3,2)
  (6,3,3,2,1)
  (6,3,2,2,1,1)
  (6,3,2,1,1,1,1)

Links

Crossrefs

The subset case is A196724.
The maximal case is A325859.
The integer partition case is A325856.
The strict integer partition case is A325855.
Cf. A002033, A108917, A143823, A325853, A325854, A325857, A325858, A325877.

Programs

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

A325855 Number of strict integer partitions of n such that every pair of distinct parts has a different product.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 14, 18, 22, 25, 31, 37, 44, 53, 59, 69, 83, 100, 111, 129, 152, 173, 198, 232, 260, 302, 342, 386, 448, 498, 565, 646, 728, 819, 918, 1039, 1164, 1310, 1462, 1631, 1830, 2053, 2282, 2532, 2825, 3136, 3482, 3869, 4300, 4744
Offset: 0

Views

Author

Gus Wiseman, May 31 2019

Keywords

Examples

			The a(1) = 1 through a(10) = 10 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)
                                                          (4321)

Links

Crossrefs

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, A325854, A325858, A325876, A325877.

Programs

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

A325857 Number of integer partitions of n such that every orderless pair of distinct parts has a different sum.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 41, 55, 74, 97, 125, 165, 209, 269, 335, 428, 527, 664, 804, 1005, 1210, 1496, 1780, 2186, 2586, 3148, 3698, 4473, 5226, 6279, 7290, 8706, 10067, 11950, 13744, 16242, 18605, 21864, 24942, 29184, 33188, 38651, 43782, 50791, 57402, 66300, 74683, 86026, 96658
Offset: 0

Views

Author

Gus Wiseman, May 31 2019

Keywords

Examples

			The A000041(14) - a(14) = 10 partitions of 14 not satisfying the condition are:
  (6,5,2,1)
  (6,4,3,1)
  (5,4,3,2)
  (5,4,2,2,1)
  (4,4,3,2,1)
  (5,4,2,1,1,1)
  (4,3,3,2,1,1)
  (4,3,2,2,2,1)
  (4,3,2,2,1,1,1)
  (4,3,2,1,1,1,1,1)

Links

Crossrefs

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, A103300, A108917, A143823, A196724, A325853, A325855, A325858, A325859, A325862.

Programs

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

Extensions

Terms a(31) onward from Max Alekseyev, Sep 23 2023

A325880 Number of maximal subsets of {1..n} containing n such that every ordered pair of distinct elements has a different difference.

Original entry on oeis.org

1, 1, 2, 2, 4, 8, 8, 10, 18, 34, 50, 70, 78, 89, 120, 181, 277, 401, 561, 728, 867, 1031, 1219, 1537, 2013, 2684, 3581, 4973, 6435, 8124, 9974, 12054, 14057, 16890, 19783, 24102, 29539, 37247, 46301, 59825, 74556, 94064, 115057, 141068, 167521, 200790, 232798, 273734
Offset: 1

Views

Author

Gus Wiseman, Jun 02 2019

Keywords

Comments

Also the number of maximal subsets of {1..n} containing n such that every orderless pair of (not necessarily distinct) elements has a different sum.

Examples

			The a(2) = 1 through a(9) = 18 subsets:
  {1,2}  {1,3}  {1,2,4}  {1,2,5}  {1,2,6}  {2,3,7}    {3,5,8}    {4,6,9}
         {2,3}  {1,3,4}  {1,4,5}  {1,3,6}  {2,4,7}    {4,5,8}    {5,6,9}
                         {2,3,5}  {1,4,6}  {2,6,7}    {1,2,4,8}  {1,2,4,9}
                         {2,4,5}  {1,5,6}  {3,4,7}    {1,2,6,8}  {1,2,6,9}
                                  {2,3,6}  {4,5,7}    {1,3,4,8}  {1,2,7,9}
                                  {2,5,6}  {4,6,7}    {1,3,7,8}  {1,3,4,9}
                                  {3,4,6}  {1,2,5,7}  {1,5,6,8}  {1,3,8,9}
                                  {3,5,6}  {1,3,6,7}  {1,5,7,8}  {1,4,8,9}
                                                      {2,3,6,8}  {1,6,7,9}
                                                      {2,4,7,8}  {1,6,8,9}
                                                                 {2,3,5,9}
                                                                 {2,3,7,9}
                                                                 {2,4,5,9}
                                                                 {2,4,8,9}
                                                                 {2,6,7,9}
                                                                 {2,6,8,9}
                                                                 {3,4,7,9}
                                                                 {3,5,8,9}

Links

Crossrefs

Cf. A002033, A108917, A143823, A143824, A196723.
Cf. A325858, A325859, A325861, A325867, A325869, A325879, A325992.

Programs

  • Mathematica
    fasmax[y_]:=Complement[y,Union@@(Most[Subsets[#]]&/@y)];
Table[Length[fasmax[Select[Subsets[Range[n]],MemberQ[#,n]&&UnsameQ@@Subtract@@@Subsets[Union[#],{2}]&]]],{n,0,10}]
  • PARI
    a(n)={
  my(ismaxl(b,w)=for(k=1, n, if(!bittest(b,k) && !bitand(w,bitor(b,1<= n, ismaxl(b,w),
         my(s=self()(k+1, b,w));
         b+=1<Andrew Howroyd, Mar 23 2025

Extensions

a(25) onwards from Andrew Howroyd, Mar 23 2025
Showing 1-10 of 14 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.