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 1-4 of 4 results.

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

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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

A094708 Size of the smallest set hitting all {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

0, 0, 1, 1, 1, 1, 2, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5, 5, 6, 6, 7, 7, 7, 7, 8, 8, 9, 9, 9, 9, 10, 10, 11, 11, 11, 11, 12, 12, 13, 13, 13, 13, 13, 14, 14, 15, 15, 15, 16, 16, 16, 17, 17, 18, 18, 18, 18, 19, 19, 19, 19, 20, 21
Offset: 1

Views

Author

Barry Cipra, Jun 15 2004

Keywords

Comments

A057561(n) = n - a(n). [Steven Finch, Feb 25 2009]

Links

Crossrefs

Cf. A003586, A057561.

Extensions

More terms from Sean A. Irvine, Nov 19 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 *)

A157282 Maximum cardinality of a weakly triple-free subset of {1, 2, ..., n}.

Original entry on oeis.org

1, 2, 2, 3, 4, 5, 6, 7, 7, 8, 9, 10, 11, 12, 12, 13, 14, 14, 15, 16, 16, 17, 18, 19, 20, 21, 21, 22, 23, 24, 25, 26, 26, 27, 28, 29, 30, 31, 31, 32, 33, 34, 35, 36, 36, 37, 38, 39, 40, 41, 41, 42, 43, 43, 44, 45, 45, 46, 47, 48, 49, 50, 50, 51, 52, 53, 54, 55, 55, 56, 57, 58, 59
Offset: 1

Views

Author

Steven Finch, Feb 26 2009

Keywords

Comments

A050294 is different from this sequence. A050294 involves sets encompassing no {x,3x}; this sequence involves sets encompassing no {x,2x,3x}.
From Steven Finch, Feb 27 2009: (Start)
Define d(n)=A003586(n), b(0)=0 and b(k)=A057561(n) for d(n) <= k < d(n+1).
Then a(n) = Sum_{m=1..ceiling(n/3)} b(floor(n/e(m))) where e(m) = A007310(m). (End)

Examples

			a(9)=7 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 upper triangles are removed by marking 2, 3.

Links

Crossrefs

A050296 is the strongly triple-free analog of this sequence.
Cf. A003586, A007310, A057561.

Extensions

More terms from Steven Finch, Feb 27 2009
Showing 1-4 of 4 results.

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

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