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 1-3 of 3 results.

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

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(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.

Links

Crossrefs

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.

Programs

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

Extensions

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

A324757 Number of integer partitions of n not containing 1 or any prime indices of the parts.

Original entry on oeis.org

1, 0, 1, 1, 2, 1, 4, 3, 4, 6, 9, 7, 14, 12, 19, 21, 28, 29, 41, 45, 56, 64, 81, 89, 114, 125, 154, 176, 211, 236, 288, 324, 383, 432, 514, 578, 678, 766, 891, 1006, 1176, 1306, 1525, 1711, 1966, 2212, 2538, 2839, 3258, 3646, 4150, 4647, 5288, 5891, 6698, 7472
Offset: 0

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

Crossrefs

The subset version is A324742, with maximal case A324763. The strict case is A324752. The Heinz number version is A324761. An infinite version is A324695.
Cf. A000720, A000837, A001462, A051424, A112798, A276625, A290822, A304360, A306844, A324764, A324742, A324753, A324756.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],!MemberQ[#,1]&&Intersection[#,PrimePi/@First/@Join@@FactorInteger/@#]=={}&]],{n,0,30}]

A324761 Heinz numbers of integer partitions not containing 1 or any prime indices of the parts.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 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, 113, 115, 121, 123, 125, 127, 129, 131, 133, 137, 139, 143, 147, 149
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}
  41: {13}
  43: {14}

Crossrefs

The subset version is A324742, with maximal case A324763. The strict integer partition version is A324752. The integer partition version is A324757. An infinite version is A324695.
Cf. A000720, A001221, A007097, A056239, A112798, A276625, A289509, A290822, A304360, A306844, A324743, A324751, A324756, A324758, A324764.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1,100,2],Intersection[primeMS[#],Union@@primeMS/@primeMS[#]]=={}&]
Showing 1-3 of 3 results.

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