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).

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, 15, 16, 16, 17, 17, 18, 18, 19, 19, 20, 20, 21, 21, 22, 22, 23, 23, 24, 24, 25, 25, 26, 26, 27, 27, 28, 28, 29, 29, 30, 30, 31, 32, 32, 33, 33, 33, 34, 34, 35, 35

Offset

1,3

Comments

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

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.

Links

Giovanni Resta, Table of n, a(n) for n = 1..1000

J. Cassaigne and P. Zimmermann, Numerical Evaluation of the Strongly Triple-Free Constant.

Julien Cassaigne and Paul Zimmermann, Numerical Evaluation of the Strongly Triple-Free Constant (pdf file, 1996).

Steven R. Finch, Triple-Free Sets of Integers [From Steven Finch, Apr 20 2019]

Example

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.

Mathematica

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 *)

Crossrefs

Cf. A057561, A094708.

Sequence in context: A358854 A330015 A331163 * A025162 A330027 A373074

Adjacent sequences: A157268 A157269 A157270 * A157272 A157273 A157274

Keyword

nonn

Author

Steven Finch, Feb 26 2009

Status

approved