cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A057561 Size of the largest set encompassing no {x, 2x, 3x} contained in D(n) = the first n 3-smooth numbers {1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, ...} (A003586).

Original entry on oeis.org

1, 2, 2, 3, 4, 5, 5, 6, 7, 7, 8, 8, 9, 10, 11, 11, 12, 13, 13, 14, 14, 15, 16, 17, 17, 18, 18, 19, 20, 21, 21, 22, 22, 23, 24, 25, 25, 26, 26, 27, 28, 29, 30, 30, 31, 31, 32, 33, 33, 34, 35, 35, 36, 36, 37, 38, 39, 39, 40, 41, 42, 42, 42, 43, 44, 45, 45, 46, 47
Offset: 1

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Author

N. J. A. Sloane

Keywords

Comments

This is the weakly triple-free analog of A157271.
A094708(n) = n - a(n).
Position of first n gives A004059(n).
Graham paper erroneously has a(30)=20. - Sean A. Irvine, Nov 18 2015

Examples

			A set for a(30) is {1, 2, 6, 8, 9, 12, 16, 27, 36, 48, 54, 64, 72, 96, 128, 162, 216, 243, 256, 288, 324}. - _Sean A. Irvine_, Oct 26 2015

References

  • R. L. Graham et al., On extremal density theorems for linear forms, pp. 103-109 of H. Zassenhaus, editor, Number Theory and Algebra. Academic Press, NY, 1977.

Links

Crossrefs

Cf. A004059, A094708, A003586, A157271.

Extensions

Edited by Steven Finch, Feb 25 2009
Revised by N. J. A. Sloane, Jun 13 2012
a(30) corrected and more terms from Sean A. Irvine, Oct 26 2015

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).

Original entry on oeis.org

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

Views

Author

Steven Finch, Feb 26 2009

Keywords

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.

Examples

			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.

Links

Crossrefs

Cf. A057561, A094708.

Programs

  • 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 *)

A004059 a(n) gives position of first n in A057561.

Original entry on oeis.org

1, 2, 4, 5, 6, 8, 9, 11, 13, 14, 15, 17, 18, 20, 22, 23, 24, 26, 28, 29, 30, 32, 34, 35, 36, 38, 40, 41, 42, 43, 45, 47, 48, 50, 51, 53, 55, 56, 57, 59, 60, 61, 64
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

References

  • R. L. Graham et al., On extremal density theorems for linear forms, pp. 103-109 of H. Zassenhaus, editor, Number Theory and Algebra. Academic Press, NY, 1977.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A057561, A094708.

Extensions

Edited by Steven Finch, Feb 25 2009
Revised by N. J. A. Sloane, Jun 13 2012
a(21) corrected and more terms from Sean A. Irvine, Nov 18 2015
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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