OFFSET
1,3
COMMENTS
Computed using the following integer programming formulation, where the decision variable x[i] is 1 if i is a member of the strongly triple-free subset, 0 otherwise. Maximize sum {i in 1..n} x[i] subject to x[i] + x[3i] <= 1 for i in 1..n such that 3i in 1..n, x[i] + x[2i] <= 1 for i in 1..n such that 2i in 1..n, x[i] in {0,1} for i in 1..n. - Rob Pratt.
The problem can also be thought of as finding a maximum independent set in a graph with nodes 1..n and edges of the form (i,3i) and (i,2i). - Rob Pratt.
LINKS
Steven R. Finch, Triple-Free Sets of Integers [From Steven Finch, Apr 20 2019]
Eric Weisstein's World of Mathematics, Free Set.
EXAMPLE
a(9)=6 since there are three grid graphs, two with a single vertex {7}, {5} and the other with rows {1,3,9}, {2,6}, {4}, {8}. The adjacencies are eliminated by marking 2, 3, 8. [From Steven Finch, Feb 26 2009]
MATHEMATICA
e[m_]:=(6*m+(-1)^m-3)/2 f[k_, n_, m_]:=1+Floor[FullSimplify[Log[n/3^k/e[m]]/Log[2]]] g[n_, m_]:=Floor[FullSimplify[Log[n/e[m]]/Log[3]]] peven[n_, m_]:=Sum[Quotient[f[k, n, m]+Mod[k+1, 2], 2], {k, 0, g[n, m]}] podd[n_, m_]:=Sum[Quotient[f[k, n, m]+Mod[k, 2], 2], {k, 0, g[n, m]}] p[n_]:=Sum[Max[peven[n, m], podd[n, m]], {m, 1, Ceiling[n/3]}] Table[p[n], {n, 1, 71}] [From Steven Finch, Feb 26 2009]
CROSSREFS
KEYWORD
nonn
AUTHOR
EXTENSIONS
More terms from Rob Pratt, Oct 25 2002
STATUS
approved