A050296 Maximum cardinality of a strongly triple-free subset of {1, 2, ..., n}.

1, 1, 2, 2, 3, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 10, 11, 11, 12, 12, 13, 13, 14, 15, 16, 16, 16, 16, 17, 18, 19, 20, 21, 21, 22, 22, 23, 23, 24, 24, 25, 26, 27, 27, 28, 28, 29, 30, 31, 31, 32, 32, 33, 33, 34, 34, 35, 35, 36, 36, 37, 37, 38, 39, 40, 41, 42, 42, 43, 43, 44

Offset: 1

Author

Eric W. Weisstein

Keywords

• nonn

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, Oct 25 2002

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, Oct 25 2002

Examples

			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. - _Steven Finch_, Feb 26 2009
		

Links

• Steven R. Finch, Triple-Free Sets of Integers. - Steven Finch, Apr 20 2019
• Eric Weisstein's World of Mathematics, Triple-Free Set.

Crossrefs

Cf. A050291-A050295.

A157282 is the weakly triple-free analog of this sequence. - Steven Finch, Feb 26 2009

Programs

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}] (* Steven Finch, Feb 26 2009 *)

Extensions

More terms from Rob Pratt, Oct 25 2002