A324737 -id:A324737 - cp's OEIS frontend

A324736 Number of subsets of {1...n} containing all prime indices of the elements.

1, 2, 3, 4, 7, 9, 15, 22, 43, 79, 127, 175, 343, 511, 851, 1571, 3141, 4397, 8765, 13147, 25243, 46843, 76795, 115171, 230299, 454939, 758203, 1516363, 2916079, 4356079, 8676079, 12132079, 24264157, 45000157, 73800253, 145685053, 291369853, 437054653, 728424421

Offset: 0

Gus Wiseman, Mar 13 2019

nonn

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.

Also the number of subsets of {1...n} containing no prime indices of the non-elements up to n.

			The a(0) = 1 through a(6) = 15 subsets:
  {}  {}   {}     {}       {}         {}           {}
      {1}  {1}    {1}      {1}        {1}          {1}
           {1,2}  {1,2}    {1,2}      {1,2}        {1,2}
                  {1,2,3}  {1,4}      {1,4}        {1,4}
                           {1,2,3}    {1,2,3}      {1,2,3}
                           {1,2,4}    {1,2,4}      {1,2,4}
                           {1,2,3,4}  {1,2,3,4}    {1,2,6}
                                      {1,2,3,5}    {1,2,3,4}
                                      {1,2,3,4,5}  {1,2,3,5}
                                                   {1,2,3,6}
                                                   {1,2,4,6}
                                                   {1,2,3,4,5}
                                                   {1,2,3,4,6}
                                                   {1,2,3,5,6}
                                                   {1,2,3,4,5,6}
An example for n = 18 is {1,2,4,7,8,9,12,16,17,18}, whose elements have the following prime indices:
   1: {}
   2: {1}
   4: {1,1}
   7: {4}
   8: {1,1,1}
   9: {2,2}
  12: {1,1,2}
  16: {1,1,1,1}
  17: {7}
  18: {1,2,2}
All of these prime indices {1,2,4,7} belong to the subset, as required.

Andrew Howroyd, Table of n, a(n) for n = 0..100

The strict integer partition version is A324748. The integer partition version is A324753. The Heinz number version is A290822. An infinite version is A324698.
Cf. A000720, A001462, A007097, A076078, A084422, A085945, A112798, A276625, A279861, A290689, A304360, A320426.
Cf. A324697, A324737, A324741, A324743.

Table[Length[Select[Subsets[Range[n]],SubsetQ[#,PrimePi/@First/@Join@@FactorInteger/@DeleteCases[#,1]]&]],{n,0,10}]

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 15 2019

Terms a(21) and beyond from Andrew Howroyd, Aug 15 2019

A324753 Number of integer partitions of n containing all prime indices of their parts.

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 5, 7, 8, 14, 16, 23, 29, 40, 49, 66, 81, 109, 133, 172, 211, 274, 332, 419, 511, 640, 775, 965, 1165, 1434, 1730, 2109, 2530, 3083, 3683, 4447, 5308, 6375, 7573, 9062, 10730, 12786, 15104, 17909, 21095, 24937, 29284, 34488, 40421, 47450

Offset: 0

Gus Wiseman, Mar 16 2019

nonn

These could be described as transitive integer partitions.

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 a(1) = 1 through a(8) = 8 integer partitions:
  (1)  (11)  (21)   (211)   (41)     (321)     (421)      (3221)
             (111)  (1111)  (221)    (411)     (2221)     (4211)
                            (2111)   (2211)    (3211)     (22211)
                            (11111)  (21111)   (4111)     (32111)
                                     (111111)  (22111)    (41111)
                                               (211111)   (221111)
                                               (1111111)  (2111111)
                                                          (11111111)

The subset version is A324736. The strict case is A324748. The Heinz number version is A290822. An infinite version is A324698.
Cf. A000720, A000837, A001462, A007097, A051424, A112798, A276625, A279861, A290689, A290760.
Cf. A324697, A324737.

Table[Length[Select[IntegerPartitions[n],SubsetQ[#,PrimePi/@First/@Join@@FactorInteger/@DeleteCases[#,1]]&]],{n,0,30}]

A324742 Number of subsets of {2...n} containing no prime indices of the elements.

Original entry on oeis.org

1, 2, 3, 6, 10, 16, 24, 48, 84, 144, 228, 420, 648, 1080, 1800, 3600, 5760, 11136, 16704, 31104, 53568, 90624, 136896, 269952, 515712, 862080, 1708800, 3171840, 4832640, 9325440, 14890752, 29781504, 52245504, 88418304, 166017024, 331628544, 497645568, 829409280

Offset: 1

Gus Wiseman, Mar 15 2019

nonn

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 a(1) = 1 through a(6) = 16 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}    {3,4}
                       {2,4,5}  {3,6}
                                {4,5}
                                {4,6}
                                {5,6}
                                {2,4,5}
                                {3,4,6}
                                {4,5,6}
An example for n = 20 is {4,5,6,12,17,18,19}, with prime indices:
   4: {1,1}
   5: {3}
   6: {1,2}
  12: {1,1,2}
  17: {7}
  18: {1,2,2}
  19: {8}
None of these prime indices {1,2,3,7,8} belong to the set, as required.

Andrew Howroyd, Table of n, a(n) for n = 1..100

The maximal case is A324763. The version for subsets of {1...n} is A324741. The strict integer partition version is A324752. The integer partition version is A324757. The Heinz number version is A324761. An infinite version is A304360.
Cf. A000720, A001462, A007097, A076078, A084422, A085945, A112798, A276625, A290689, A290822, A306844, A324764.
Cf. A324695, A324737, A324743, A324751, A324756, A324758.

Table[Length[Select[Subsets[Range[2,n]],Intersection[#,PrimePi/@First/@Join@@FactorInteger/@#]=={}&]],{n,10}]

pset(n)={my(b=0,f=factor(n)[,1]); sum(i=1, #f, 1<<(primepi(f[i])))}
a(n)={my(p=vector(n-1,k,pset(k+1)>>1), d=0); for(i=1, #p, d=bitor(d, p[i]));
((k,b)->if(k>#p, 1, my(t=self()(k+1,b)); if(!bitand(p[k], b), t+=if(bittest(d,k), self()(k+1, b+(1<Andrew Howroyd, Aug 16 2019

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

A324748 Number of strict integer partitions of n containing all prime indices of the parts.

Original entry on oeis.org

1, 1, 0, 1, 0, 1, 1, 1, 0, 2, 1, 2, 3, 2, 2, 4, 3, 4, 3, 5, 6, 9, 8, 7, 8, 11, 12, 13, 15, 17, 22, 22, 20, 28, 31, 32, 36, 41, 43, 53, 53, 59, 70, 76, 77, 89, 99, 108, 124, 135, 139, 160, 172, 188, 209, 229, 243, 274, 298, 315, 353, 391, 417, 457, 496, 538, 588

Offset: 0

Gus Wiseman, Mar 15 2019

nonn

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 first 15 terms count the following integer partitions.
   1: (1)
   3: (2,1)
   5: (4,1)
   6: (3,2,1)
   7: (4,2,1)
   9: (8,1)
   9: (6,2,1)
  10: (4,3,2,1)
  11: (8,2,1)
  11: (5,3,2,1)
  12: (9,2,1)
  12: (7,4,1)
  12: (6,3,2,1)
  13: (8,4,1)
  13: (6,4,2,1)
  14: (8,3,2,1)
  14: (7,4,2,1)
  15: (12,2,1)
  15: (9,3,2,1)
  15: (8,4,2,1)
  15: (5,4,3,2,1)
An example for n = 6 is (20,18,11,5,3,2,1), with prime indices:
  20: {1,1,3}
  18: {1,2,2}
  11: {5}
   5: {3}
   3: {2}
   2: {1}
   1: {}
All of these prime indices {1,2,3,5} belong to the partition, as required.

The subset version is A324736. The non-strict version is A324753. The Heinz number version is A290822. An infinite version is A324698.
Cf. A000720, A001462, A007097, A074971, A078374, A112798, A276625, A279861, A290689, A290760, A305713.
Cf. A324697, A324737, A324751.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&SubsetQ[#,PrimePi/@First/@Join@@FactorInteger/@DeleteCases[#,1]]&]],{n,0,30}]

A324750 Number of strict 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, 1, 1, 2, 3, 2, 4, 4, 4, 6, 8, 8, 11, 10, 15, 16, 19, 23, 27, 28, 35, 39, 47, 50, 63, 68, 77, 91, 102, 114, 130, 147, 169, 187, 213, 237, 268, 300, 336, 380, 422, 472, 525, 587, 647, 731, 810, 895, 996, 1102, 1227, 1355, 1498, 1661, 1818, 2020, 2221

Offset: 0

Gus Wiseman, Mar 15 2019

nonn

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 a(2) = 1 through a(17) = 15 strict integer partitions (A...H = 10...17):
  2  3  4  5  6   7   8   9   A   B    C    D    E    F    G    H
              42  43  62  54  64  65   75   76   86   87   97   98
                  52      63  73  83   84   85   95   96   A6   A7
                          72  82  542  93   94   A4   A5   C4   B6
                                       A2   A3   B3   B4   D3   C5
                                       642  B2   C2   C3   E2   D4
                                            643  752  D2   763  E3
                                            652  842  654  862  F2
                                                      762  943  854
                                                      843  A42  863
                                                      852       872
                                                                A43
                                                                A52
                                                                B42
                                                                6542

The subset version is A324739. The non-strict version is A324755. The Heinz number version is A324760. An infinite version is A324694.
Cf. A000720, A001462, A007097, A074971, A078374, A112798, A276625, A290822, A304360, A305713, A306844.
Cf. A324696, A324737, A324742, A324744, A324764.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&!MemberQ[#,1]&&!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

Gus Wiseman, Mar 17 2019

nonn

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).

			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}

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.

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]]]&]

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

Gus Wiseman, Mar 14 2019

nonn

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 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}

Andrew Howroyd, Table of n, a(n) for n = 1..100

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.

Table[Length[Select[Subsets[Range[2,n]],!MemberQ[#,k_/;SubsetQ[#,PrimePi/@First/@FactorInteger[k]]]&]],{n,10}]

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

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

cp's OEIS Frontend

A324736 Number of subsets of {1...n} containing all prime indices of the elements.

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A324753 Number of integer partitions of n containing all prime indices of their parts.

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A324742 Number of subsets of {2...n} containing no prime indices of the elements.

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A324748 Number of strict integer partitions of n containing all prime indices of the parts.

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A324750 Number of strict integer partitions of n not containing 1 or any part whose prime indices all belong to the partition.

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

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

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