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A174094
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Number of ways to choose n positive integers less than or equal to 2n such that none of the n integers divides another.
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5
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1, 2, 2, 3, 5, 4, 6, 12, 10, 14, 26, 26, 34, 68, 48, 72, 120, 120, 168, 336, 264, 396, 792, 624, 816, 1632, 1632, 2208, 3616, 3616, 5056, 10112, 6592, 9888, 19776, 19776, 24384, 48768, 48768, 73152, 112320, 76032, 114048, 228096, 190080, 264960, 529920
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OFFSET
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0,2
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COMMENTS
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REFERENCES
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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.
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LINKS
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C. Bindi, L. Bussoli, M. Grazzini, M. Pellegrini, G. Pirillo, M. A. Pirillo, and A. Troise, Su un risultato di uno studente di Erdös, Periodico di matematiche 1 (2016), Vol 8, No 1 (2016): Serie XII Anno CXXVI. [Broken link]
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EXAMPLE
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a(1) = 2 because we can choose {1}, {2}.
a(2) = 2 because we can choose {2, 3}, {3, 4}.
a(3) = 3 because we can choose {2, 3, 5}, {3, 4, 5}, {4, 5, 6}.
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MAPLE
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F:= proc(S, m)
option remember;
local s, S1, S2;
if nops(S) < m then return 0 fi;
if m = 1 then return nops(S) fi;
s:= min(S);
S1:= S minus {s};
S2:= S minus {seq(j*s, j=1..floor(max(S)/s))};
F(S1, m) + F(S2, m-1);
end proc;
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MATHEMATICA
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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]];
a[n_] := F[Range[2n], n];
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CROSSREFS
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The smallest n integers possible is A174063.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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