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 12 results. Next

A325858 Number of Golomb partitions of n.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 10, 14, 20, 25, 36, 47, 59, 78, 99, 122, 155, 195, 232, 295, 355, 432, 522, 641, 749, 919, 1076, 1283, 1506, 1802, 2067, 2470, 2835, 3322, 3815, 4496, 5070, 5959, 6736, 7807, 8849, 10266, 11499, 13326, 14928, 17140, 19193, 22037, 24519, 28106
Offset: 0

Views

Author

Gus Wiseman, Jun 02 2019

Keywords

Comments

We define a Golomb partition of n to be an integer partition of n such that every pair of distinct parts has a different difference.
Also the number of integer partitions of n such that every orderless pair of (not necessarily distinct) parts has a different sum.
The strict case is A325876.

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)   (411)     (331)
                            (11111)  (2211)    (421)
                                     (3111)    (511)
                                     (21111)   (2221)
                                     (111111)  (4111)
                                               (22111)
                                               (31111)
                                               (211111)
                                               (1111111)
The A000041(9) - a(9) = 5 non-Golomb partitions of 9 are: (531), (432), (3321), (32211), (321111).

Links

Crossrefs

The subset case is A143823.
The maximal case is A325879.
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, A325325, A325853, A325856, A325868.

Programs

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

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

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

Cf. A007865, A051026, A054519, A067992, A103580, A325853, A325854, A325860, A325861, A325994, A326078.

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

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}]

A325876 Number of strict Golomb partitions of n.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 5, 6, 6, 9, 11, 10, 15, 17, 18, 24, 29, 27, 38, 43, 47, 53, 67, 67, 84, 87, 102, 113, 137, 131, 167, 179, 204, 213, 261, 263, 315, 327, 377, 413, 476, 472, 564, 602, 677, 707, 820, 845, 969, 1027, 1131, 1213, 1364, 1413, 1596, 1700, 1858
Offset: 0

Views

Author

Gus Wiseman, Jun 02 2019

Keywords

Comments

We define a Golomb partition of n to be an integer partition of n such that every ordered pair of distinct parts has a different difference.
Also the number of strict integer partitions of n such that every orderless pair of (not necessarily distinct) parts has a different sum.
The non-strict case is A325858.

Examples

			The a(2) = 1 through a(11) = 11 partitions (A = 10, B = 11):
  (2)  (3)   (4)   (5)   (6)   (7)    (8)    (9)    (A)    (B)
       (21)  (31)  (32)  (42)  (43)   (53)   (54)   (64)   (65)
                   (41)  (51)  (52)   (62)   (63)   (73)   (74)
                               (61)   (71)   (72)   (82)   (83)
                               (421)  (431)  (81)   (91)   (92)
                                      (521)  (621)  (532)  (A1)
                                                    (541)  (542)
                                                    (631)  (632)
                                                    (721)  (641)
                                                           (731)
                                                           (821)

Links

Crossrefs

The subset case is A143823.
The maximal case is A325879.
The integer partition case is A325858.
The strict integer partition case is A325876.
Heinz numbers of the counterexamples are given by A325992.
Cf. A002033, A275972, A325325, A325853, A325856, A325868.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&UnsameQ@@Subtract@@@Subsets[Union[#],{2}]&]],{n,0,30}]
  • Python
    from collections import Counter
from itertools import combinations
from sympy.utilities.iterables import partitions
def A325876(n): return sum(1 for p in partitions(n) if max(list(Counter(abs(d[0]-d[1]) for d in combinations(list(Counter(p).elements()),2)).values()),default=1)==1)-(n&1^1) if n else 1 # Chai Wah Wu, Sep 17 2023

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}]

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

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
Showing 1-10 of 12 results. Next

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