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A324736 Number of subsets of {1...n} containing all prime indices of the elements. 13
1, 2, 3, 4, 7, 9, 15, 22, 43, 79, 127, 175, 343, 511, 851, 1571, 3141, 4397, 8765, 13147, 25243 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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

LINKS

Table of n, a(n) for n=0..20.

EXAMPLE

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.

MATHEMATICA

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

CROSSREFS

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, A324740, A324741, A324743.

Sequence in context: A276846 A303665 A027947 * A293853 A067236 A320357

Adjacent sequences:  A324733 A324734 A324735 * A324737 A324738 A324739

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, Mar 13 2019

STATUS

approved

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Last modified June 20 16:26 EDT 2019. Contains 324234 sequences. (Running on oeis4.)