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A324737 Number of subsets of {2...n} containing every element of {2...n} whose prime indices all belong to the subset. 8
1, 2, 3, 6, 8, 16, 24, 48, 84, 168, 216, 432, 648, 1296, 2448, 4896, 6528, 13056, 19584, 39168, 77760, 155520, 229248, 458496, 790272, 1580544, 3128832, 6257664, 9386496, 18772992, 24081408, 48162816, 95938560, 191877120, 378335232, 756670464, 1135005696, 2270011392 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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.

Also the number of subsets of {2...n} with complement containing no term whose prime indices all belong to the subset.

LINKS

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

EXAMPLE

The a(1) = 1 through a(6) = 16 subsets:

  {}  {}   {}     {}       {}         {}

      {2}  {3}    {3}      {4}        {4}

           {2,3}  {4}      {5}        {5}

                  {2,3}    {3,5}      {6}

                  {3,4}    {4,5}      {3,5}

                  {2,3,4}  {2,3,5}    {4,5}

                           {3,4,5}    {4,6}

                           {2,3,4,5}  {5,6}

                                      {2,3,5}

                                      {3,4,5}

                                      {3,5,6}

                                      {4,5,6}

                                      {2,3,4,5}

                                      {2,3,5,6}

                                      {3,4,5,6}

                                      {2,3,4,5,6}

An example for n = 15 is {2, 3, 5, 8, 9, 10, 11, 15}. The numbers from 2 to 15 with all prime indices in the subset are {3, 5, 9, 11, 15}, which all belong to the subset, as required.

MATHEMATICA

Table[Length[Select[Subsets[Range[2, n]], Function[set, SubsetQ[set, Select[Range[2, n], SubsetQ[set, PrimePi/@First/@FactorInteger[#]]&]]]]], {n, 10}]

PROG

(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+(1<<k))); if(bitnegimply(p[k], b), t+=if(bittest(d, k), self()(k+1, b), t)); t))(1, 0)} \\ Andrew Howroyd, Aug 24 2019

CROSSREFS

Cf. A000720, A001221, A001462, A007097, A084422, A085945, A112798, A276625, A290689, A290822, A304360, A306844.

Cf. A324697, A324698, A324736, A324738, A324748, A324753, A324755.

Sequence in context: A029867 A056348 A308546 * A057574 A198296 A276033

Adjacent sequences:  A324734 A324735 A324736 * A324738 A324739 A324740

KEYWORD

nonn

AUTHOR

Gus Wiseman, Mar 13 2019

EXTENSIONS

Terms a(21) and beyond from Andrew Howroyd, Aug 24 2019

STATUS

approved

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Last modified February 19 09:33 EST 2020. Contains 332041 sequences. (Running on oeis4.)