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}.
		

Crossrefs

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.

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}
		

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.

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

A326023 Number of subsets of {1..n} containing all of their integer quotients.

Original entry on oeis.org

1, 2, 3, 5, 9, 17, 25, 49, 73, 145, 217, 433, 553, 1105, 1657, 2593, 3937, 7873, 10057, 20113, 26689, 42321, 63481, 126961, 154801, 309601, 464401, 737569, 992161, 1984321, 2450881, 4901761, 6292801, 10197313, 15295969, 26241697, 32947489, 65894977, 98842465, 161587873, 205842529
Offset: 0

Views

Author

Gus Wiseman, Jun 04 2019

Keywords

Comments

These are sets that are closed under taking the quotient of two (not necessarily distinct) divisible terms.

Examples

			The a(0) = 1 through a(5) = 17 subsets:
  {}  {}   {}     {}       {}         {}
      {1}  {1}    {1}      {1}        {1}
           {1,2}  {1,2}    {1,2}      {1,2}
                  {1,3}    {1,3}      {1,3}
                  {1,2,3}  {1,4}      {1,4}
                           {1,2,3}    {1,5}
                           {1,2,4}    {1,2,3}
                           {1,3,4}    {1,2,4}
                           {1,2,3,4}  {1,2,5}
                                      {1,3,4}
                                      {1,3,5}
                                      {1,4,5}
                                      {1,2,3,4}
                                      {1,2,3,5}
                                      {1,2,4,5}
                                      {1,3,4,5}
                                      {1,2,3,4,5}
		

Crossrefs

Programs

  • Mathematica
    Table[Length[Select[Subsets[Range[n]],SubsetQ[#,Select[Divide@@@Tuples[#,2],IntegerQ]]&]],{n,0,10}]

Formula

For n > 0, a(n) = A326078(n) + 1.

Extensions

Terms a(21) and beyond from Andrew Howroyd, Aug 30 2019

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}
		

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.

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.

Programs

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

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.

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.

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).
		

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.

Programs

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

A326077 Number of maximal primitive subsets of {1..n}.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 4, 4, 6, 7, 11, 11, 13, 13, 23, 24, 36, 36, 48, 48, 64, 66, 126, 126, 150, 151, 295, 363, 507, 507, 595, 595, 895, 903, 1787, 1788, 2076, 2076, 4132, 4148, 5396, 5396, 6644, 6644, 9740, 11172, 22300, 22300, 26140, 26141, 40733, 40773, 60333, 60333, 80781, 80783
Offset: 0

Views

Author

Gus Wiseman, Jun 05 2019

Keywords

Comments

a(n) is the number of maximal primitive subsets of {1, ..., n}. Here primitive means that no element of the subset divides any other and maximal means that no element can be added to the subset while maintaining the property of being pairwise indivisible. - Nathan McNew, Aug 10 2020

Examples

			The a(0) = 1 through a(9) = 7 sets:
  {}  {1}  {1}  {1}   {1}   {1}    {1}    {1}     {1}     {1}
           {2}  {23}  {23}  {235}  {235}  {2357}  {2357}  {2357}
                      {34}  {345}  {345}  {3457}  {3457}  {2579}
                                   {456}  {4567}  {3578}  {3457}
                                                  {4567}  {3578}
                                                  {5678}  {45679}
                                                          {56789}
		

Crossrefs

Programs

  • Mathematica
    stableQ[u_, Q_]:=!Apply[Or, Outer[#1=!=#2&&Q[#1, #2]&, u, u, 1], {0, 1}];
    fasmax[y_]:=Complement[y, Union@@(Most[Subsets[#]]&/@y)];
    Table[Length[fasmax[Select[Subsets[Range[n]],stableQ[#,Divisible]&]]],{n,0,10}]
  • PARI
    divset(n)={sumdiv(n, d, if(dif(k>#p, ismax(b), my(f=!bitand(p[k], b)); if(!f || bittest(d, k), self()(k+1, b)) + if(f, self()(k+1, b+(1<Andrew Howroyd, Aug 30 2019

Extensions

Terms a(19) to a(55) from Andrew Howroyd, Aug 30 2019
Name edited by Nathan McNew, Aug 10 2020

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}
		

Crossrefs

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
Showing 1-10 of 14 results. Next