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 1-9 of 9 results.

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

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

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

Views

Author

Gus Wiseman, May 31 2019

Keywords

Examples

			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}

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, A325858, A325860, A325861, A325869, A325878, A325879, A325880.

Programs

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

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}]

A325853 Number of integer partitions of n such that every pair of distinct parts has a different quotient.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 14, 21, 28, 39, 51, 69, 88, 116, 148, 193, 242, 309, 385, 484, 596, 746, 915, 1128, 1371, 1679, 2030, 2460, 2964, 3570, 4268, 5115, 6088, 7251, 8584, 10175, 12002, 14159, 16619, 19526, 22846, 26713, 31153, 36300, 42169, 48990, 56728
Offset: 0

Views

Author

Gus Wiseman, May 31 2019

Keywords

Comments

Also the number of integer partitions of n such that every orderless pair of (not necessarily distinct) parts has a different product.

Examples

			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)   (321)     (331)
                            (11111)  (411)     (511)
                                     (2211)    (2221)
                                     (3111)    (3211)
                                     (21111)   (4111)
                                     (111111)  (22111)
                                               (31111)
                                               (211111)
                                               (1111111)
The one partition of 7 for which not every pair of distinct parts has a different quotient is (4,2,1).

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, A108917, A143823, A196724, A325768, A325856, A325868, A325869, A325876.

Programs

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

A325994 Heinz numbers of integer partitions such that not every ordered pair of distinct parts has a different quotient.

Original entry on oeis.org

42, 84, 126, 168, 210, 230, 252, 294, 336, 378, 390, 399, 420, 460, 462, 504, 546, 588, 630, 672, 690, 714, 742, 756, 780, 798, 840, 882, 920, 924, 966, 1008, 1050, 1092, 1134, 1150, 1170, 1176, 1197, 1218, 1260, 1302, 1344, 1365, 1380, 1386, 1428, 1470, 1484
Offset: 1

Views

Author

Gus Wiseman, Jun 02 2019

Keywords

Comments

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

Examples

			The sequence of terms together with their prime indices begins:
    42: {1,2,4}
    84: {1,1,2,4}
   126: {1,2,2,4}
   168: {1,1,1,2,4}
   210: {1,2,3,4}
   230: {1,3,9}
   252: {1,1,2,2,4}
   294: {1,2,4,4}
   336: {1,1,1,1,2,4}
   378: {1,2,2,2,4}
   390: {1,2,3,6}
   399: {2,4,8}
   420: {1,1,2,3,4}
   460: {1,1,3,9}
   462: {1,2,4,5}
   504: {1,1,1,2,2,4}
   546: {1,2,4,6}
   588: {1,1,2,4,4}
   630: {1,2,2,3,4}
   672: {1,1,1,1,1,2,4}

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, A056239, A103300, A108917, A112798, A143823, A196724, A325768, A325856, A325868, A325869, A325876.

Programs

  • Mathematica
    Select[Range[1000],!UnsameQ@@Divide@@@Subsets[PrimePi/@First/@FactorInteger[#],{2}]&]

A325854 Number of strict integer partitions of n such that every pair of distinct parts has a different quotient.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 4, 6, 8, 9, 12, 13, 16, 20, 23, 30, 33, 41, 47, 52, 61, 75, 90, 98, 116, 132, 151, 173, 206, 226, 263, 297, 337, 387, 427, 488, 555, 623, 697, 782, 886, 984, 1108, 1240, 1374, 1545, 1726, 1910, 2120, 2358, 2614, 2903, 3218, 3567, 3933
Offset: 0

Views

Author

Gus Wiseman, May 31 2019

Keywords

Comments

Also the number of strict integer partitions of n such that every pair of (not necessarily distinct) parts has a different product.

Examples

			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)
                                           (431)  (81)   (91)
                                           (521)  (432)  (532)
                                                  (531)  (541)
                                                  (621)  (631)
                                                         (721)
The two strict partitions of 13 such that not every pair of distinct parts has a different quotient are (9,3,1) and (6,4,2,1).

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. A108917, A143823, A196724, A275972, A325768, A325855, A325858, A325868, A325869, A325876, A325877.

Programs

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

A325868 Number of subsets of {1..n} containing n such that every ordered pair of distinct elements has a different quotient.

Original entry on oeis.org

1, 2, 4, 6, 14, 24, 52, 84, 120, 240, 548, 688, 1784, 2600, 4236, 5796, 16200, 17568, 49968, 55648, 101360, 176792, 433736, 430032, 728784, 1360928, 2304840, 2990856, 8682912, 7877376, 25243200, 27946656, 46758912, 81457248, 121546416, 114388320, 442583952
Offset: 1

Views

Author

Gus Wiseman, Jun 02 2019

Keywords

Examples

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

Links

Crossrefs

Cf. A025582, A108917, A143823, A196723, A196724.
Cf. A325853, A325854, A325858, A325860, A325861, A325864, A325869.

Programs

  • Mathematica
    Table[Length[Select[Subsets[Range[n]],MemberQ[#,n]&&UnsameQ@@Divide@@@Subsets[#,{2}]&]],{n,10}]

Extensions

a(21)-a(37) from Fausto A. C. Cariboni, Oct 16 2020

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 1-9 of 9 results.

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