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A157271 Size of the largest set encompassing no {x, 2x} nor {x, 3x} contained in D(n) = the first n 3-smooth numbers {1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27,...} (A003586). 2

%I #16 Apr 20 2019 11:52:34

%S 1,1,2,2,3,3,4,4,5,5,6,6,7,7,8,8,9,9,10,11,11,11,12,12,13,13,14,14,15,

%T 15,16,16,17,17,18,18,19,19,20,20,21,21,22,22,23,23,24,24,25,25,26,26,

%U 27,27,28,28,29,29,30,30,31,32,32,33,33,33,34,34,35,35

%N Size of the largest set encompassing no {x, 2x} nor {x, 3x} contained in D(n) = the first n 3-smooth numbers {1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27,...} (A003586).

%C This is the strongly triple-free analog of A057561 and the description is modeled after A094708.

%C a(n) is the size of the maximal independent set in a grid graph with vertex set D(n) and edges connecting every x to 2x and every x to 3x.

%H Giovanni Resta, <a href="/A157271/b157271.txt">Table of n, a(n) for n = 1..1000</a>

%H J. Cassaigne and P. Zimmermann, <a href="http://www.people.fas.harvard.edu/~sfinch/csolve/triple/cz.pdf">Numerical Evaluation of the Strongly Triple-Free Constant</a>.

%H Julien Cassaigne and Paul Zimmermann, <a href="https://oeis.org/A086316/a086316_3.pdf">Numerical Evaluation of the Strongly Triple-Free Constant</a> (pdf file, 1996).

%H Steven R. Finch, <a href="/FinchTriple.html">Triple-Free Sets of Integers</a> [From Steven Finch, Apr 20 2019]

%e For n=7, the grid graph has rows {1,3,9}, {2,6}, {4}, {8} and the maximal set of nonadjacent vertices is {1,4,6,9}, hence a(7)=4.

%t f[k_,n_]:=1+Floor[FullSimplify[Log[n/3^k]/Log[2]]]; g[n_]:=Floor[FullSimplify[Log[n]/Log[3]]]; peven[n_]:=Sum[Quotient[f[k,n]+Mod[k+1,2],2],{k,0,g[n]}]; podd[n_]:=Sum[Quotient[f[k,n]+Mod[k,2],2],{k,0,g[n]}]; p[n_]:=Max[peven[n],podd[n]]; v[1]=1;j=1;k=1;n=70; For[k=2, k<=n, k++, If[2*v[k-j]<3^j,v[k]=2*v[k-j],{v[k]=3^j,j++}]]; Table[p[v[n]],{n,1,70}] (* _Steven Finch_, Feb 27 2009; corrected by _Giovanni Resta_, Jul 29 2015 *)

%Y Cf. A057561, A094708.

%K nonn

%O 1,3

%A _Steven Finch_, Feb 26 2009

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