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 1-8 of 8 results.

A324738 Number of subsets of {1...n} containing no element > 1 whose prime indices all belong to the subset.

Original entry on oeis.org

1, 2, 3, 5, 8, 13, 26, 42, 72, 120, 232, 376, 752, 1128, 2256, 4512, 8256, 13632, 27264, 42048, 82944, 158976, 313344, 497664, 995328, 1700352, 3350016, 5815296, 11630592, 17491968, 34983936, 56954880, 108933120, 210788352, 418258944, 804667392, 1609334784
Offset: 0

Views

Author

Gus Wiseman, Mar 13 2019

Keywords

Comments

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

Examples

			The a(0) = 1 through a(6) = 26 subsets:
  {}  {}   {}   {}     {}     {}       {}
      {1}  {1}  {1}    {1}    {1}      {1}
           {2}  {2}    {2}    {2}      {2}
                {3}    {3}    {3}      {3}
                {1,3}  {4}    {4}      {4}
                       {1,3}  {5}      {5}
                       {2,4}  {1,3}    {6}
                       {3,4}  {1,5}    {1,3}
                              {2,4}    {1,5}
                              {2,5}    {1,6}
                              {3,4}    {2,4}
                              {4,5}    {2,5}
                              {2,4,5}  {2,6}
                                       {3,4}
                                       {3,6}
                                       {4,5}
                                       {4,6}
                                       {5,6}
                                       {1,3,6}
                                       {1,5,6}
                                       {2,4,5}
                                       {2,4,6}
                                       {2,5,6}
                                       {3,4,6}
                                       {4,5,6}
                                       {2,4,5,6}

Links

Crossrefs

The maximal case is A324744. The case of subsets of {2...n} is A324739. The strict integer partition version is A324749. The integer partition version is A324754. The Heinz number version is A324759. An infinite version is A324694.
Cf. A000720, A001221, A001462, A007097, A076078, A084422, A085945, A112798, A276625, A279861, A290689, A290822, A304360, A306844.
Cf. A324695, A324736, A324741, A324750, A324755, A324760.

Programs

  • Mathematica
    Table[Length[Select[Subsets[Range[n]],!MemberQ[#,k_/;SubsetQ[#,PrimePi/@First/@FactorInteger[k]]]&]],{n,0,10}]
  • PARI
    pset(n)={my(b=0,f=factor(n)[,1]); sum(i=1, #f, 1<<(primepi(f[i])))}
a(n)={my(p=vector(n,k,if(k==1, 1, pset(k))), d=0); for(i=1, #p, d=bitor(d, p[i]));
((k,b)->if(k>#p, 1, my(t=self()(k+1,b)); if(bitnegimply(p[k], b), t+=if(bittest(d,k), self()(k+1, b+(1<Andrew Howroyd, Aug 16 2019

Extensions

Terms a(21) and beyond from Andrew Howroyd, Aug 16 2019

A324755 Number of integer partitions of n not containing 1 or any part whose prime indices all belong to the partition.

Original entry on oeis.org

1, 0, 1, 1, 2, 1, 4, 3, 5, 6, 10, 7, 16, 14, 23, 23, 35, 34, 53, 54, 75, 80, 112, 115, 160, 169, 223, 244, 315, 339, 442, 478, 604, 664, 832, 910, 1131, 1245, 1524, 1689, 2054, 2263, 2743, 3039, 3634, 4042, 4809, 5343, 6326, 7035, 8276, 9217, 10795, 12011
Offset: 0

Views

Author

Gus Wiseman, Mar 16 2019

Keywords

Comments

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
For example, (6,2) is such a partition because the prime indices of 6 are {1,2}, which do not all belong to the partition. On the other hand, (5,3) is not such a partition because the prime indices of 5 are {3}, and 3 belongs to the partition.

Examples

			The a(2) = 1 through a(10) = 10 integer partitions (A = 10):
  (2)  (3)  (4)   (5)  (6)    (7)   (8)     (9)    (A)
            (22)       (33)   (43)  (44)    (54)   (55)
                       (42)   (52)  (62)    (63)   (64)
                       (222)        (422)   (72)   (73)
                                    (2222)  (333)  (82)
                                            (522)  (433)
                                                   (442)
                                                   (622)
                                                   (4222)
                                                   (22222)

Crossrefs

The subset version is A324739, with maximal case A324762. The strict case is A324750. The Heinz number version is A324760. An infinite version is A324694.
Cf. A000837, A001462, A051424, A112798, A276625, A290822, A304360, A306844, A324695, A324696, A324744.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],!MemberQ[#,k_/;SubsetQ[#,PrimePi/@First/@If[k==1,{},FactorInteger[k]]]]&]],{n,0,30}]

A324759 Heinz numbers of integer partitions containing no part > 1 whose prime indices all belong to the partition.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 16, 17, 19, 20, 21, 22, 23, 25, 26, 27, 29, 31, 32, 33, 34, 35, 37, 39, 40, 41, 43, 44, 46, 47, 49, 50, 51, 52, 53, 57, 58, 59, 61, 62, 63, 64, 65, 67, 68, 71, 73, 74, 77, 79, 80, 81, 82, 83, 85, 86, 87, 88, 89, 91, 92, 93
Offset: 1

Views

Author

Gus Wiseman, Mar 17 2019

Keywords

Comments

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. 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:
   1: {}
   2: {1}
   3: {2}
   4: {1,1}
   5: {3}
   7: {4}
   8: {1,1,1}
   9: {2,2}
  10: {1,3}
  11: {5}
  13: {6}
  16: {1,1,1,1}
  17: {7}
  19: {8}
  20: {1,1,3}
  21: {2,4}
  22: {1,5}
  23: {9}
  25: {3,3}
  26: {1,6}

Crossrefs

The subset version is A324738, with maximal case A324744. The strict integer partition version is A324749. The integer partition version is A324754. An infinite version is A324694.
Cf. A000720, A001221, A007097, A056239, A112798, A276625, A289509, A290822, A306844, A324695, A324750, A324755, A324760.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],!MemberQ[DeleteCases[primeMS[#],1],k_/;SubsetQ[primeMS[#],primeMS[k]]]&]

A324754 Number of integer partitions of n containing no part > 1 whose prime indices all belong to the partition.

Original entry on oeis.org

1, 1, 2, 2, 4, 3, 7, 8, 11, 12, 19, 19, 30, 34, 46, 50, 71, 76, 104, 119, 151, 171, 225, 247, 315, 360, 446, 504, 629, 703, 867, 986, 1192, 1346, 1636, 1837, 2204, 2500, 2965, 3348, 3980, 4475, 5276, 5963, 6973, 7852, 9194, 10335, 12009, 13536, 15650, 17589
Offset: 0

Views

Author

Gus Wiseman, Mar 16 2019

Keywords

Comments

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
For example, (6,2) is such a partition because the prime indices of 6 are {1,2}, which do not all belong to the partition. On the other hand, (5,3) is not such a partition because the prime indices of 5 are {3}, and 3 belongs to the partition.

Examples

			The a(1) = 1 through a(8) = 11  integer partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (311)    (33)      (43)       (44)
                    (31)    (11111)  (42)      (52)       (62)
                    (1111)           (51)      (61)       (71)
                                     (222)     (331)      (422)
                                     (3111)    (511)      (611)
                                     (111111)  (31111)    (2222)
                                               (1111111)  (3311)
                                                          (5111)
                                                          (311111)
                                                          (11111111)

Crossrefs

The subset version is A324738, with maximal case A324744. The strict case is A324749. The Heinz number version is A324759. An infinite version is A324694.
Cf. A000837, A001462, A007097, A051424, A112798, A276625, A290822, A304360, A306844, A324695, A324750, A324755, A324760.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],!MemberQ[#,k_/;SubsetQ[#,PrimePi/@First/@FactorInteger[k]]]&]],{n,0,30}]

A324760 Heinz numbers of integer partitions not containing 1 or any part whose prime indices all belong to the partition.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 47, 49, 51, 53, 57, 59, 61, 63, 65, 67, 71, 73, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 107, 109, 111, 113, 115, 117, 121, 123, 125, 127, 129, 131, 133, 137, 139
Offset: 1

Views

Author

Gus Wiseman, Mar 17 2019

Keywords

Comments

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. 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:
   1: {}
   3: {2}
   5: {3}
   7: {4}
   9: {2,2}
  11: {5}
  13: {6}
  17: {7}
  19: {8}
  21: {2,4}
  23: {9}
  25: {3,3}
  27: {2,2,2}
  29: {10}
  31: {11}
  33: {2,5}
  35: {3,4}
  37: {12}
  39: {2,6}
  41: {13}

Crossrefs

The subset version is A324739, with maximal case A324762. The strict integer partition version is A324750. The integer partition version is A324755. An infinite version is A324694.
Cf. A000720, A001221, A007097, A056239, A112798, A289509, A290822, A306844, A324695, A324696, A324737, A324744.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],!MemberQ[primeMS[#],k_/;SubsetQ[primeMS[#],primeMS[k]]]&]

A324762 Number of maximal subsets of {2...n} containing no element whose prime indices all belong to the subset.

Original entry on oeis.org

1, 1, 2, 2, 2, 2, 4, 4, 6, 6, 8, 8, 16, 16, 16, 16, 16, 16, 32, 32, 40, 40, 52, 52, 64, 64, 72, 72, 144, 144, 176, 176, 200, 200, 232, 232, 464, 464, 464, 464, 536, 536, 1072, 1072, 1072, 1072, 2144, 2144, 2400, 2400, 2400, 2400, 4800, 4800, 4800, 4800, 4800
Offset: 1

Views

Author

Gus Wiseman, Mar 17 2019

Keywords

Comments

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

Examples

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

Links

Crossrefs

The non-maximal version is A324739.
The version for subsets of {1...n} is A324744.
An infinite version is A324694.
Cf. A076078, A084422, A085945, A112798, A276625, A304360, A306844.
Cf. A324695, A324738, A324742, A324743, A324763, A324750, A324754, A324755, A324760.

Programs

  • Mathematica
    maxim[s_]:=Complement[s,Last/@Select[Tuples[s,2],UnsameQ@@#&&SubsetQ@@#&]];
Table[Length[maxim[Select[Subsets[Range[2,n]],!MemberQ[#,k_/;SubsetQ[#,PrimePi/@First/@FactorInteger[k]]]&]]],{n,10}]
  • PARI
    pset(n)={my(b=0, f=factor(n)[, 1]); sum(i=1, #f, 1<<(primepi(f[i])))}
a(n)={my(p=vector(n, k, pset(k)), d=0); for(i=1, #p, d=bitor(d, p[i]));
my(ismax(b)=for(k=1, #p, if(!bittest(b,k) && bitnegimply(p[k], b), my(e=bitor(b, 1<#p, ismax(b), my(f=bitnegimply(p[k], b)); if(!f || bittest(d, k), self()(k+1, b)) + if(f, self()(k+1, b+(1<Andrew Howroyd, Aug 27 2019

Extensions

Terms a(16) and beyond from Andrew Howroyd, Aug 27 2019

A324739 Number of subsets of {2...n} containing no element whose prime indices all belong to the subset.

Original entry on oeis.org

1, 2, 3, 6, 10, 20, 30, 60, 96, 192, 312, 624, 936, 1872, 3744, 7488, 12480, 24960, 37440, 74880, 142848, 285696, 456192, 912384, 1548288, 3096576, 5308416, 10616832, 15925248, 31850496, 51978240, 103956480, 200835072, 401670144, 771489792, 1542979584, 2314469376
Offset: 1

Views

Author

Gus Wiseman, Mar 14 2019

Keywords

Comments

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

Examples

			The a(1) = 1 through a(6) = 20 subsets:
  {}  {}   {}   {}     {}       {}
      {2}  {2}  {2}    {2}      {2}
           {3}  {3}    {3}      {3}
                {4}    {4}      {4}
                {2,4}  {5}      {5}
                {3,4}  {2,4}    {6}
                       {2,5}    {2,4}
                       {3,4}    {2,5}
                       {4,5}    {2,6}
                       {2,4,5}  {3,4}
                                {3,6}
                                {4,5}
                                {4,6}
                                {5,6}
                                {2,4,5}
                                {2,4,6}
                                {2,5,6}
                                {3,4,6}
                                {4,5,6}
                                {2,4,5,6}

Links

Crossrefs

The maximal case is A324762. The case of subsets of {1...n} is A324738. The strict integer partition version is A324750. The integer partition version is A324755. The Heinz number version is A324760. An infinite version is A324694.
Cf. A000720, A001221, A001462, A007097, A084422, A085945, A112798, A276625, A279861, A290689, A290822, A304360, A306844.
Cf. A324695, A324696, A324736, A324737, A324741, A324742, A324744, A324764.

Programs

  • Mathematica
    Table[Length[Select[Subsets[Range[2,n]],!MemberQ[#,k_/;SubsetQ[#,PrimePi/@First/@FactorInteger[k]]]&]],{n,10}]
  • PARI
    pset(n)={my(b=0,f=factor(n)[,1]); sum(i=1, #f, 1<<(primepi(f[i])))}
a(n)={my(p=vector(n,k,pset(k)), d=0); for(i=1, #p, d=bitor(d, p[i]));
((k,b)->if(k>#p, 1, my(t=self()(k+1,b)); if(bitnegimply(p[k], b), t+=if(bittest(d,k), self()(k+1, b+(1<Andrew Howroyd, Aug 16 2019

Extensions

Terms a(21) and beyond from Andrew Howroyd, Aug 16 2019

A324749 Number of strict integer partitions of n containing no part > 1 whose prime indices all belong to the partition.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 4, 3, 4, 6, 6, 8, 11, 10, 14, 14, 19, 21, 26, 28, 35, 38, 44, 50, 60, 65, 79, 88, 98, 113, 131, 144, 165, 185, 211, 234, 268, 297, 334, 374, 420, 470, 525, 584, 649, 727, 801, 902, 998, 1100, 1220, 1357, 1500, 1657, 1833, 2029, 2220, 2462
Offset: 0

Views

Author

Gus Wiseman, Mar 15 2019

Keywords

Comments

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

Examples

			The a(0) = 1 through a(10) = 6 strict integer partitions:
  ()  (1)  (2)  (3)  (4)    (5)  (6)    (7)    (8)    (9)    (10)
                     (3,1)       (4,2)  (4,3)  (6,2)  (5,4)  (6,4)
                                 (5,1)  (5,2)  (7,1)  (6,3)  (7,3)
                                        (6,1)         (7,2)  (8,2)
                                                             (9,1)
                                                             (6,3,1)

Crossrefs

The subset version is A324738. The non-strict version is A324754. The Heinz number version is A324759. An infinite version is A324694.
Cf. A000720, A001462, A007097, A074971, A078374, A112798, A276625, A290822, A305713.
Cf. A324744, A324750, A324755, A324760.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&!MemberQ[#,k_/;SubsetQ[#,PrimePi/@First/@FactorInteger[k]]]&]],{n,0,30}]
Showing 1-8 of 8 results.

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