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A174094 Number of ways to choose n positive integers less than or equal to 2n such that none of the n integers divides another. 4

%I

%S 2,2,3,5,4,6,12,10,14,26,26,34,68,48,72,120,120,168,336,264,396,792,

%T 624,816,1632,1632,2208,3616,3616,5056,10112,6592,9888,19776,19776,

%U 24384,48768,48768,73152,112320,76032,114048,228096,190080,264960,529920

%N Number of ways to choose n positive integers less than or equal to 2n such that none of the n integers divides another.

%C a(n) >= 2^(1+floor((n-1)/3)). - _Robert Israel_, Aug 25 2015

%H Bertram Felgenhauer, <a href="/A174094/b174094.txt">Table of n, a(n) for n = 1..3400</a> (terms 1..100 from Marco Pellegrini)

%H C. Bindi, L. Bussoli, M. Grazzini, M. Pellegrini, G. Pirillo, M.A. Pirillo, A. Troise, <a href="http://parsec2.unina2.it/ojs/index.php/periodico/article/view/285">Su un risultato di uno studente di Erdös</a>, Periodico di matematiche 1 (2016), Vol 8, No 1 (2016): Serie XII Anno CXXVI.

%H Bertram Felgenhauer, <a href="/A174094/a174094.hs.txt">Haskell program for computing A174094</a>.

%H Hong Liu, Péter Pál Pach, Richárd Palincza, <a href="https://arxiv.org/abs/1805.06341">The number of maximum primitive sets of integers</a>, arXiv:1805.06341 [math.CO], 2018.

%H Sujith Vijay, <a href="https://arxiv.org/abs/1804.01740">On large primitive subsets of {1,2,...,2n}</a>, arXiv:1804.01740 [math.CO], 2018.

%e a(1) = 2 because we can choose {1}, {2}.

%e a(2) = 2 because we can choose {2, 3}, {3, 4}.

%e a(3) = 3 because we can choose {2, 3, 5}, {3, 4, 5}, {4, 5, 6}.

%p F:= proc(S,m)

%p option remember;

%p local s,S1,S2;

%p if nops(S) < m then return 0 fi;

%p if m = 1 then return nops(S) fi;

%p s:= min(S);

%p S1:= S minus {s};

%p S2:= S minus {seq(j*s,j=1..floor(max(S)/s))};

%p F(S1, m) + F(S2, m-1);

%p end proc;

%p seq(F({$1..2*n},n), n=1..37); # _Robert Israel_, Aug 25 2015

%t F[S_List, m_] := F[S, m] = Module[{s, S1, S2}, If[Length[S]<m, Return[0]]; If[m == 1, Return[Length[S]]]; s = Min[S]; S1 = S ~Complement~ {s}; S2 = S ~Complement~ Table[j*s, {j, 1, Floor[Max[S]/s]}]; F[S1, m] + F[S2, m - 1]];

%t a[n_] := F[Range[2n], n];

%t Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 1, 37}] (* _Jean-François Alcover_, Jul 10 2018, after _Robert Israel_ *)

%Y The smallest n integers possible is A174063.

%K nonn

%O 1,1

%A _David Brown_, Mar 07 2010

%E More terms from _David Brown_, Mar 14 2010

%E a(30)-a(46) from _Ray Chandler_, Mar 19 2010

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Last modified February 18 15:04 EST 2020. Contains 332019 sequences. (Running on oeis4.)