A324736 - OEIS

%I #16 Aug 22 2019 08:24:28

%S 1,2,3,4,7,9,15,22,43,79,127,175,343,511,851,1571,3141,4397,8765,

%T 13147,25243,46843,76795,115171,230299,454939,758203,1516363,2916079,

%U 4356079,8676079,12132079,24264157,45000157,73800253,145685053,291369853,437054653,728424421

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

%C 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.

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

%H Andrew Howroyd, <a href="/A324736/b324736.txt">Table of n, a(n) for n = 0..100</a>

%e The a(0) = 1 through a(6) = 15 subsets:

%e   {}  {}   {}     {}       {}         {}           {}

%e       {1}  {1}    {1}      {1}        {1}          {1}

%e            {1,2}  {1,2}    {1,2}      {1,2}        {1,2}

%e                   {1,2,3}  {1,4}      {1,4}        {1,4}

%e                            {1,2,3}    {1,2,3}      {1,2,3}

%e                            {1,2,4}    {1,2,4}      {1,2,4}

%e                            {1,2,3,4}  {1,2,3,4}    {1,2,6}

%e                                       {1,2,3,5}    {1,2,3,4}

%e                                       {1,2,3,4,5}  {1,2,3,5}

%e                                                    {1,2,3,6}

%e                                                    {1,2,4,6}

%e                                                    {1,2,3,4,5}

%e                                                    {1,2,3,4,6}

%e                                                    {1,2,3,5,6}

%e                                                    {1,2,3,4,5,6}

%e An example for n = 18 is {1,2,4,7,8,9,12,16,17,18}, whose elements have the following prime indices:

%e    1: {}

%e    2: {1}

%e    4: {1,1}

%e    7: {4}

%e    8: {1,1,1}

%e    9: {2,2}

%e   12: {1,1,2}

%e   16: {1,1,1,1}

%e   17: {7}

%e   18: {1,2,2}

%e All of these prime indices {1,2,4,7} belong to the subset, as required.

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

%o (PARI)

%o pset(n)={my(b=0, f=factor(n)[,1]); sum(i=1, #f, 1<<(primepi(f[i])))}

%o a(n)={my(p=vector(n,k,pset(k)), d=0); for(i=1, #p, d=bitor(d, p[i]));

%o ((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<<k)), t)); t))(1,0)} \\ _Andrew Howroyd_, Aug 15 2019

%Y The strict integer partition version is A324748. The integer partition version is A324753. The Heinz number version is A290822. An infinite version is A324698.

%Y Cf. A000720, A001462, A007097, A076078, A084422, A085945, A112798, A276625, A279861, A290689, A304360, A320426.

%Y Cf. A324697, A324737, A324741, A324743.

%K nonn

%O 0,2

%A _Gus Wiseman_, Mar 13 2019

%E Terms a(21) and beyond from _Andrew Howroyd_, Aug 15 2019