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%I #50 Jun 24 2022 17:11:55
%S 1,2,2,3,5,4,6,12,10,14,26,26,34,68,48,72,120,120,168,336,264,396,792,
%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
%D F. Caldarola, G. d'Atri, M.Pellegrini, Combinatorics on n-sets: Arithmetic properties and numerical results. In: Sergeyev Y., Kvasov D. (eds) Numerical Computations: Theory and Algorithms. NUMTA 2019. Lecture Notes in Computer Science, vol. 11973. Springer, Cham, 389-401.
%H Bertram Felgenhauer, <a href="/A174094/b174094.txt">Table of n, a(n) for n = 0..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 0,2
%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
%E a(0)=1 prepended by _Alois P. Heinz_, Jun 24 2022
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