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.

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

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

Views

Author

Gus Wiseman, Jun 02 2019

Keywords

Comments

The non-strict case is A325857.

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)
                                     (421)  (431)  (81)   (91)
                                            (521)  (432)  (532)
                                                   (531)  (541)
                                                   (621)  (631)
                                                          (721)

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, A108917, A143823, A275972.
Cf. A325854, A325855, A325858, A325862, A325864, A325876, A325879.

Programs

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

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}]

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}]

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}]

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

A325991 Heinz numbers of integer partitions such that not every orderless pair of distinct parts has a different sum.

Original entry on oeis.org

210, 420, 462, 630, 840, 858, 910, 924, 1050, 1155, 1260, 1326, 1386, 1470, 1680, 1716, 1820, 1848, 1870, 1890, 1938, 2100, 2145, 2310, 2470, 2520, 2574, 2622, 2652, 2730, 2772, 2926, 2940, 3150, 3234, 3315, 3360, 3432, 3465, 3570, 3640, 3696, 3740, 3780, 3876
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:
   210: {1,2,3,4}
   420: {1,1,2,3,4}
   462: {1,2,4,5}
   630: {1,2,2,3,4}
   840: {1,1,1,2,3,4}
   858: {1,2,5,6}
   910: {1,3,4,6}
   924: {1,1,2,4,5}
  1050: {1,2,3,3,4}
  1155: {2,3,4,5}
  1260: {1,1,2,2,3,4}
  1326: {1,2,6,7}
  1386: {1,2,2,4,5}
  1470: {1,2,3,4,4}
  1680: {1,1,1,1,2,3,4}
  1716: {1,1,2,5,6}
  1820: {1,1,3,4,6}
  1848: {1,1,1,2,4,5}
  1870: {1,3,5,7}
  1890: {1,2,2,2,3,4}

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, A056239, A103300, A108917, A112798, A143823, A196724, A325853, A325855, A325858, A325859, A325862, A325992, A325993, A325994.

Programs

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

A325993 Heinz numbers of integer partitions such that not every orderless pair of distinct parts has a different product.

Original entry on oeis.org

390, 780, 798, 1170, 1365, 1560, 1596, 1914, 1950, 2340, 2394, 2590, 2730, 2886, 3120, 3192, 3510, 3828, 3900, 3990, 4095, 4290, 4386, 4485, 4680, 4788, 5070, 5170, 5180, 5460, 5586, 5742, 5772, 5850, 6042, 6240, 6384, 6630, 6699, 6825, 7020, 7182, 7410, 7656
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:
   390: {1,2,3,6}
   780: {1,1,2,3,6}
   798: {1,2,4,8}
  1170: {1,2,2,3,6}
  1365: {2,3,4,6}
  1560: {1,1,1,2,3,6}
  1596: {1,1,2,4,8}
  1914: {1,2,5,10}
  1950: {1,2,3,3,6}
  2340: {1,1,2,2,3,6}
  2394: {1,2,2,4,8}
  2590: {1,3,4,12}
  2730: {1,2,3,4,6}
  2886: {1,2,6,12}
  3120: {1,1,1,1,2,3,6}
  3192: {1,1,1,2,4,8}
  3510: {1,2,2,2,3,6}
  3828: {1,1,2,5,10}
  3900: {1,1,2,3,3,6}
  3990: {1,2,3,4,8}

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, A056239, A108917, A112798, A143823, A292886, A325858, A325877, A325991, A325992, A325994.

Programs

  • Mathematica
    Select[Range[1000],!UnsameQ@@Times@@@Subsets[PrimePi/@First/@FactorInteger[#],{2}]&]
Showing 1-9 of 9 results.

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