OFFSET
0,2
COMMENTS
a(n) >= 2^(1+floor((n-1)/3)). - Robert Israel, Aug 25 2015
REFERENCES
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.
LINKS
Bertram Felgenhauer, Table of n, a(n) for n = 0..3400 (terms 1..100 from Marco Pellegrini)
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]
Bertram Felgenhauer, Haskell program for computing A174094.
Hong Liu, Péter Pál Pach, and Richárd Palincza, The number of maximum primitive sets of integers, arXiv:1805.06341 [math.CO], 2018.
Richárd Palincza, Counting type and extremal problems from Arithmetic Combinatorics, Ph. D. Thesis, Budapest Univ. Tech. Econ. (Hungary, 2024).
Sujith Vijay, On large primitive subsets of {1,2,...,2n}, arXiv:1804.01740 [math.CO], 2018.
EXAMPLE
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}.
MAPLE
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;
seq(F({$1..2*n}, n), n=1..37); # Robert Israel, Aug 25 2015
MATHEMATICA
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];
Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 1, 37}] (* Jean-François Alcover, Jul 10 2018, after Robert Israel *)
CROSSREFS
KEYWORD
nonn
AUTHOR
David Brown, Mar 07 2010
EXTENSIONS
More terms from David Brown, Mar 14 2010
a(30)-a(46) from Ray Chandler, Mar 19 2010
a(0)=1 prepended by Alois P. Heinz, Jun 24 2022
STATUS
approved