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.

A325365 Number of maximal subsets of {1..n} containing n such that no two elements have the same sorted prime signature.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 4, 4, 8, 4, 20, 4, 12, 12, 48, 8, 56, 16, 64, 48, 48, 36, 324, 162, 81, 567, 378, 168, 1680, 168, 1848, 264, 264, 264, 2640, 240, 288, 288, 3456, 576, 7488, 1152, 4032, 4032, 2016, 1872, 28080, 9360, 6240, 3360, 6720, 3584, 28672, 6144
Offset: 1

Views

Author

Gus Wiseman, Jul 06 2019

Keywords

Comments

The sorted prime signature (A118914) of a positive integer is the multiset of exponents in its standard factorization into prime numbers.

Examples

			The a(1) = 1 through a(12) = 20 subsets (A = 10, B = 11, C = 12) are the following. The common cardinality of sets in column n is A085089(n).
  1  12  13  124  145  1246  1467  12468  12689  1248A  1468B  12468C
             134       1346        13468  13689  1289A  148AB  1248AC
                       1456        14568  15689  1348A  1689B  12689C
                                   14678  16789  1389A  189AB  1289AC
                                                 1458A         13468C
                                                 1478A         1348AC
                                                 1589A         13689C
                                                 1789A         1389AC
                                                               14568C
                                                               1458AC
                                                               14678C
                                                               1468BC
                                                               1478AC
                                                               148ABC
                                                               15689C
                                                               1589AC
                                                               16789C
                                                               1689BC
                                                               1789AC
                                                               189ABC

Crossrefs

Cf. A001221, A001222, A025487, A064839, A085089, A112798, A118914, A124010, A181819, A324762, A325263, A325365, A326438, A326441.

Programs

  • Mathematica
    prisig[n_]:=If[n==1,{},Sort[Last/@FactorInteger[n]]];
Table[Times@@Length/@Split[Sort[Array[prisig,n]]]/Count[Array[prisig,n],prisig[n]],{n,30}]

Formula

a(n) = A326439(n)/A064839(n).

A326439 Number of maximal subsets of {1..n} such that no two elements have the same sorted prime signature.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 4, 4, 8, 16, 20, 20, 24, 36, 48, 48, 56, 112, 128, 192, 240, 288, 324, 324, 486, 567, 1134, 1512, 1680, 1680, 1848, 1848, 2112, 2376, 2640, 2640, 2880, 3168, 3456, 6912, 7488, 14976, 16128, 20160, 24192, 26208, 28080, 28080, 37440, 43680
Offset: 0

Views

Author

Gus Wiseman, Jul 06 2019

Keywords

Comments

The sorted prime signature (A118914) of a positive integer is the multiset of exponents in its standard factorization into prime numbers.

Examples

			The a(0) = 1 through a(9) = 8 subsets:
  {}  {1}  {12}  {12}  {124}  {124}  {1246}  {1246}  {12468}  {12468}
                 {13}  {134}  {134}  {1346}  {1346}  {13468}  {12689}
                              {145}  {1456}  {1456}  {14568}  {13468}
                                             {1467}  {14678}  {13689}
                                                              {14568}
                                                              {14678}
                                                              {15689}
                                                              {16789}

Links

Crossrefs

Cf. A001221, A001222, A025487, A064839, A085089, A112798, A118914, A124010, A181819, A324762, A325263, A325365, A326438, A326441.

Programs

  • Mathematica
    prisig[n_]:=If[n==1,{},Sort[Last/@FactorInteger[n]]];
Table[Times@@(Length/@Split[Sort[Array[prisig,n]]]),{n,0,30}]
  • PARI
    a(n)={if(n==0, 1, my(M=Map()); for(i=1, n, my(f=factor(i)[,2], s=sum(k=1, #f, x^f[k]), z); mapput(M, s, if(mapisdefined(M, s, &z), z + 1, 1))); vecprod(Mat(M)[,2]))} \\ Andrew Howroyd, Aug 30 2019

A325263 Number of subsets of {1..n} containing n such that no two elements have the same sorted prime signature.

Original entry on oeis.org

1, 2, 2, 6, 4, 16, 8, 40, 40, 60, 36, 216, 72, 168, 168, 840, 240, 960, 360, 1080, 864, 864, 672, 6720, 4480, 2560, 10240, 7680, 3840, 42240, 7680, 92160, 23040, 23040, 23040, 253440, 42240, 49920, 49920, 324480, 74880, 524160, 112320, 336960, 336960, 181440
Offset: 1

Views

Author

Gus Wiseman, Jul 06 2019

Keywords

Comments

The sorted prime signature (A118914) of a positive integer is the multiset of exponents in its standard factorization into prime numbers.

Examples

			The a(1) = 1 through a(5) = 16 subsets:
  {1}  {2}    {3}    {4}      {5}      {6}        {7}
       {1,2}  {1,3}  {1,4}    {1,5}    {1,6}      {1,7}
                     {2,4}    {4,5}    {2,6}      {4,7}
                     {3,4}    {1,4,5}  {3,6}      {6,7}
                     {1,2,4}           {4,6}      {1,4,7}
                     {1,3,4}           {5,6}      {1,6,7}
                                       {1,2,6}    {4,6,7}
                                       {1,3,6}    {1,4,6,7}
                                       {1,4,6}
                                       {1,5,6}
                                       {2,4,6}
                                       {3,4,6}
                                       {4,5,6}
                                       {1,2,4,6}
                                       {1,3,4,6}
                                       {1,4,5,6}

Crossrefs

Cf. A001221, A001222, A025487, A064839, A085089, A112798, A118914, A124010, A181819, A325365, A326438, A326439, A326441.

Programs

  • Mathematica
    prisig[n_]:=If[n==1,{},Sort[Last/@FactorInteger[n]]];
Table[Times@@(1+Length/@Split[Sort[Array[prisig,n]]])/(1+Count[Array[prisig,n],prisig[n]]),{n,30}]

Formula

a(n) = A326438(n)/(1 + A064839(n)).

A326438 Number of subsets of {1..n} such that no two elements have the same sorted prime signature.

Original entry on oeis.org

1, 2, 4, 6, 12, 16, 32, 40, 80, 120, 180, 216, 432, 504, 672, 840, 1680, 1920, 2880, 3240, 4320, 5184, 6048, 6720, 13440, 17920, 20480, 30720, 38400, 42240, 84480, 92160, 184320, 207360, 230400, 253440, 506880, 549120, 599040, 648960, 973440
Offset: 0

Views

Author

Gus Wiseman, Jul 06 2019

Keywords

Comments

The sorted prime signature (A118914) of a positive integer is the multiset of exponents in its standard factorization into prime numbers.

Examples

			The a(0) = 1 through a(5) = 16 subsets:
  {}  {}   {}     {}     {}       {}
      {1}  {1}    {1}    {1}      {1}
           {2}    {2}    {2}      {2}
           {1,2}  {3}    {3}      {3}
                  {1,2}  {4}      {4}
                  {1,3}  {1,2}    {5}
                         {1,3}    {1,2}
                         {1,4}    {1,3}
                         {2,4}    {1,4}
                         {3,4}    {1,5}
                         {1,2,4}  {2,4}
                         {1,3,4}  {3,4}
                                  {4,5}
                                  {1,2,4}
                                  {1,3,4}
                                  {1,4,5}

Crossrefs

Cf. A001221, A001222, A025487, A064839, A085089, A112798, A118914, A124010, A181819, A325263, A325365, A326439, A326441.

Programs

  • Mathematica
    prisig[n_]:=If[n==1,{},Sort[Last/@FactorInteger[n]]];
Table[Length[Select[Subsets[Range[n]],UnsameQ@@prisig/@#&]],{n,0,10}]

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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