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 1-5 of 5 results.

A326156 Number of nonempty subsets of {1..n} whose product is divisible by their sum.

Original entry on oeis.org

0, 1, 2, 4, 5, 10, 19, 34, 64, 129, 267, 541, 1104, 2253, 4694, 9804, 18894, 38539, 76063, 155241, 311938, 636120, 1299869, 2653853, 5183363, 10272289, 20958448, 40945577, 81745769, 167048919, 329598054, 671038751, 1301431524, 2618590422, 5305742557, 10582105199, 20660489585, 42075929255, 85443680451, 172057673225, 338513788818
Offset: 0

Views

Author

Gus Wiseman, Jun 11 2019

Keywords

Examples

			The a(1) = 1 through a(6) = 19 subsets:
  {1}  {1}  {1}      {1}      {1}          {1}
       {2}  {2}      {2}      {2}          {2}
            {3}      {3}      {3}          {3}
            {1,2,3}  {4}      {4}          {4}
                     {1,2,3}  {5}          {5}
                              {1,2,3}      {6}
                              {1,4,5}      {3,6}
                              {2,3,5}      {1,2,3}
                              {3,4,5}      {1,4,5}
                              {1,2,3,4,5}  {2,3,5}
                                           {2,4,6}
                                           {3,4,5}
                                           {4,5,6}
                                           {1,2,3,6}
                                           {1,3,5,6}
                                           {3,4,5,6}
                                           {1,2,3,4,5}
                                           {1,2,3,4,6}
                                           {2,3,4,5,6}

Links

Crossrefs

Cf. A053632, A057567, A057568, A059529, A063865, A301987, A326150, A326151, A326153/A326154, A326155, A326158, A326178, A326179, A326180.

Programs

  • Mathematica
    Table[Length[Select[Subsets[Range[n],{1,n}],Divisible[Times@@#,Plus@@#]&]],{n,0,10}]

Extensions

a(21)-a(30) from Alois P. Heinz, Jun 13 2019
a(31)-a(40) from Bert Dobbelaere, Jun 22 2019

A326178 Number of subsets of {1..n} whose product is equal to their sum.

Original entry on oeis.org

0, 1, 2, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67
Offset: 0

Views

Author

Gus Wiseman, Jun 13 2019

Keywords

Comments

Same as A001477 (the nonnegative integers) with 3 removed.

Examples

			The a(1) = 1 through a(9) = 10 subsets:
  {1}  {1}  {1}      {1}      {1}      {1}      {1}      {1}      {1}
       {2}  {2}      {2}      {2}      {2}      {2}      {2}      {2}
            {3}      {3}      {3}      {3}      {3}      {3}      {3}
            {1,2,3}  {4}      {4}      {4}      {4}      {4}      {4}
                     {1,2,3}  {5}      {5}      {5}      {5}      {5}
                              {1,2,3}  {6}      {6}      {6}      {6}
                                       {1,2,3}  {7}      {7}      {7}
                                                {1,2,3}  {8}      {8}
                                                         {1,2,3}  {9}
                                                                  {1,2,3}

Crossrefs

Cf. A053632, A057567, A057568, A063865, A301987, A326156, A326158, A326179, A326180.

Programs

  • Mathematica
    Table[Length[Select[Subsets[Range[n]],Times@@#==Plus@@#&]],{n,0,10}]

A326441 Number of subsets of {1..n} whose sum is equal to the product of their complement.

Original entry on oeis.org

0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 3, 1, 2, 1, 3, 3, 3, 3, 1, 4, 4, 3, 2, 2, 4, 3, 5, 3, 2, 4, 5, 4, 5, 6, 1, 4, 2, 5, 4, 7, 4, 4, 3, 3, 6, 14, 3, 4, 10, 6, 3, 6, 4, 4, 4, 8, 7, 6, 8, 7, 10, 5, 11, 8, 5, 11, 4, 7, 7, 5, 8, 12, 5, 6, 9, 8, 11, 8, 5, 8, 9, 8, 10, 8
Offset: 0

Views

Author

Gus Wiseman, Jul 07 2019

Keywords

Comments

Essentially the same as A178830. - R. J. Mathar, Jul 12 2019

Examples

			The initial terms count the following subsets:
   1: {1}
   3: {1,2}
   5: {3,5}
   6: {3,4,5}
   7: {2,4,5,7}
   8: {2,4,5,6,7}
   9: {2,3,5,6,7,9}
  10: {4,5,6,8,9,10}
  10: {2,3,5,6,7,8,9}
  10: {1,2,3,4,5,8,9,10}
Also the number of subsets of {1..n} whose product is equal to the sum of their complement. For example, the initial terms count the following subsets:
   1: {}
   3: {3}
   5: {1,2,4}
   6: {1,2,6}
   7: {1,3,6}
   8: {1,3,8}
   9: {1,4,8}
  10: {6,7}
  10: {1,4,10}
  10: {1,2,3,7}

Links

Crossrefs

Cf. A028422, A053632, A059529, A063865, A178830, A301987, A325044, A325538, A326172, A326173, A326174, A326175, A326179, A326180.

Programs

  • Maple
    b:= proc(n, s, p)
      `if`(s=p, 1, `if`(n<1, 0, b(n-1, s, p)+
      `if`(s-n b(n, n*(n+1)/2, 1):
seq(a(n), n=0..100);  # Alois P. Heinz, Jul 12 2019
  • Mathematica
    Table[Length[Select[Subsets[Range[n]],Plus@@#==Times@@Complement[Range[n],#]&]],{n,0,10}]

Extensions

a(21)-a(83) from Giovanni Resta, Jul 08 2019

A326179 Number of subsets of {1..n} containing n whose product is divisible by their sum.

Original entry on oeis.org

0, 1, 1, 2, 1, 5, 9, 15, 30, 65, 138, 274, 563, 1149, 2441, 5110, 9090, 19645, 37524, 79178, 156697, 324182, 663749, 1353984, 2529510, 5088926, 10686159, 19987129, 40800192, 85303150, 162549135, 341440697, 630392773, 1317158898, 2687152135, 5276362642, 10078384386, 21415439670, 43367751196, 86613992774, 166456115593
Offset: 0

Views

Author

Gus Wiseman, Jun 13 2019

Keywords

Examples

			The a(1) = 1 through a(7) = 15 subsets:
  {1}  {2}  {3}      {4}  {5}          {6}          {7}
            {1,2,3}       {1,4,5}      {3,6}        {1,6,7}
                          {2,3,5}      {2,4,6}      {2,5,7}
                          {3,4,5}      {4,5,6}      {3,4,7}
                          {1,2,3,4,5}  {1,2,3,6}    {3,5,7}
                                       {1,3,5,6}    {1,2,4,7}
                                       {3,4,5,6}    {2,3,6,7}
                                       {1,2,3,4,6}  {2,5,6,7}
                                       {2,3,4,5,6}  {3,5,6,7}
                                                    {1,2,5,6,7}
                                                    {1,3,4,5,7}
                                                    {1,3,4,6,7}
                                                    {2,3,4,5,7}
                                                    {2,4,5,6,7}
                                                    {1,2,3,4,5,6,7}

Crossrefs

Cf. A053632, A057567, A057568, A059529, A063865, A301987, A326153/A326154, A326156, A326158, A326178, A326180.

Programs

  • Mathematica
    Table[Length[Select[Subsets[Range[n],{1,n}],MemberQ[#,n]&&Divisible[Times@@#,Plus@@#]&]],{n,0,10}]

Extensions

a(21)-a(30) from Alois P. Heinz, Jun 13 2019
a(31)-a(40) from Bert Dobbelaere, Jun 23 2019

A325538 Number of subsets of {1..n} whose product is one more than the sum of their complement.

Original entry on oeis.org

1, 1, 1, 1, 1, 0, 0, 1, 0, 0, 1, 0, 2, 0, 2, 1, 2, 1, 1, 2, 1, 2, 2, 1, 2, 0, 4, 2, 2, 4, 2, 2, 5, 5, 1, 2, 2, 3, 2, 5, 3, 4, 2, 2, 3, 10, 2, 4, 7, 5, 3, 3, 7, 6, 4, 4, 5, 5, 5, 2, 6, 4, 6, 5, 3, 8, 4, 5, 4, 5, 2, 10, 5, 3, 7, 11, 6, 10, 5, 11, 6, 4, 7, 6, 10
Offset: 0

Views

Author

Gus Wiseman, Jul 07 2019

Keywords

Comments

Also by definition the number of subsets whose sum is one fewer than the product of their complement.

Examples

			The initial terms count the following subsets:
   0: {}
   1: {1}
   2: {2}
   3: {1,3}
   4: {2,3}
   7: {4,5}
  10: {1,6,7}
  12: {7,9}
  12: {1,2,4,8}
  14: {2,5,9}
  14: {1,2,4,11}
  15: {1,3,5,7}
  16: {3,4,10}
  16: {1,3,5,8}
  17: {1,10,13}
  18: {2,5,15}
  19: {11,15}
  19: {1,2,6,14}
  20: {1,4,6,8}

Links

Crossrefs

Cf. A028422, A053632, A059529, A063865, A178830, A301987, A325041, A326172, A326173, A326174, A326175, A326179, A326180, A326441.

Programs

  • Mathematica
    Table[Length[Select[Subsets[Range[n]],1+Plus@@#==Times@@Complement[Range[n],#]&]],{n,0,10}]
ric[n_, pr_, s_, lst_, t_] := Block[{k}, If[pr == t-s, cnt++]; Do[ If[pr k <= t, ric[n, pr k, s + k, k, t], Break[]], {k, lst+1, n}]]; a[n_] := (cnt = 0; ric[n, 1, 0, 0, n (n + 1)/2 + 1]; cnt); a /@ Range[0, 85] (* Giovanni Resta, Sep 13 2019 *)

Extensions

More terms from Alois P. Heinz, Jul 12 2019
Showing 1-5 of 5 results.

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