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 1-7 of 7 results.

A085071 Integers reached in A085068.

Original entry on oeis.org

0, 4, 4, 4, 8, 84, 8, 84, 20, 12, 84, 20, 16, 24, 84, 20, 40, 56, 24, 84, 36, 28, 40, 56, 32, 148, 84, 36, 68, 52, 40, 56, 104, 44, 148, 84, 48, 120, 68, 52, 72, 132, 56, 104, 452, 60, 148, 84, 64, 88, 120, 68, 168, 404, 72, 132, 100, 76, 104, 452, 80, 196, 148, 84, 872, 116, 88
Offset: 0

Views

Author

N. J. A. Sloane, Aug 11 2003

Keywords

Links

  • J. C. Lagarias and N. J. A. Sloane, Approximate squaring (pdf, ps), Experimental Math., 13 (2004), 113-128.

Programs

  • Maple
    a:= proc(n) local i; n; for i do 4/3*ceil(%);
      if %::integer then return % fi od
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Mar 01 2021
  • Mathematica
    a[n_] := Module[{k = 4n/3}, While[!IntegerQ[k], k = 4* Ceiling[k]/3]; k];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Feb 17 2023 *)
  • Python3
    from fractions import Fraction
def A085071(n):
    c, x = 0, Fraction(n,1)
    while x.denominator > 1 or x <= n:
        x = Fraction(4*x._ceil_(),3)
        c += 1
    return x.numerator # Chai Wah Wu, Mar 01 2021

A085328 Record values in A085068.

Original entry on oeis.org

1, 3, 9, 15, 17, 18, 24, 27, 28, 30, 40, 41, 44, 47, 48, 50, 51, 52, 53, 56, 57, 60, 64, 67
Offset: 1

Views

Author

N. J. A. Sloane, Aug 13 2003

Keywords

Links

  • J. C. Lagarias and N. J. A. Sloane, Approximate squaring (pdf, ps), Experimental Math., 13 (2004), 113-128.

Crossrefs

Cf. A085068, A085071, A085330.

Programs

  • Mathematica
    f[n_] := Block[{c = 1, k = 4 n/3}, While[ ! IntegerQ@k, c++; k = 4 Ceiling@k/3]; c]; t = Table[0, {1000}]; Do[ a = f@n; If[a < 101 && t[[a]] == 0, t[[a]] = n; Print[{a, n}]], {n, 0, 21*10^8}] (* Robert G. Wilson v, May 28 2007 *)

Extensions

More terms from Jason Earls, Aug 14 2003
a(12)-a(16) from Robert G. Wilson v, May 28 2007
a(17)-a(24) from Lars Blomberg, Mar 03 2018

A085330 Where records occur in A085068.

Original entry on oeis.org

0, 1, 5, 161, 1772, 3097, 3473, 23084, 38752, 335165, 491729, 38248700, 49050536, 95305397, 1019659805, 1549919921, 2973640172, 4527000701, 6300121204, 10663850980, 30980417576, 40783699961, 57033894608, 409565230433
Offset: 1

Views

Author

N. J. A. Sloane, Aug 13 2003

Keywords

Links

  • J. C. Lagarias and N. J. A. Sloane, Approximate squaring (pdf, ps), Experimental Math., 13 (2004), 113-128.

Crossrefs

Cf. A085068, A085071, A085328.

Programs

  • Mathematica
    f[n_] := Block[{c = 1, k = 4 n/3}, While[ ! IntegerQ@k, c++; k = 4 Ceiling@k/3]; c]; t = Table[0, {1000}]; Do[ a = f@n; If[a < 101 && t[[a]] == 0, t[[a]] = n; Print[{a, n}]], {n, 0, 21*10^8}] (* Robert G. Wilson v, May 28 2007 *)
  • PARI
    f(n)=(4/3)*ceil(n); { a(n) = local(c, k); c=1; k=n; while(frac(f(k))!=0, k=f(k); c++); c } { rec=0; for(n=0,5*10^5,if(a(n)>rec,rec=a(n); print1(n":"a(n)","))) }

Extensions

More terms from Jason Earls, Aug 14 2003
a(12)-a(16) from Robert G. Wilson v, May 28 2007
a(17)-a(24) from Lars Blomberg, Mar 03 2018

A129377 First occurrence of n in A085068.

Original entry on oeis.org

3, 2, 1, 17, 19, 14, 10, 7, 5, 500, 404, 311, 233, 215, 161, 2363, 1772, 3097, 11474, 8605, 8234, 6175, 4631, 3473, 34196, 30779, 23084, 38752, 422549, 335165, 1500443, 1125332, 2039653, 2762863, 2072147, 1554110, 1165582, 874186, 655639, 491729
Offset: 1

Views

Author

Robert G. Wilson v, May 27 2007

Keywords

Comments

A085068: Number of steps for iteration of map x -> (4/3)*ceiling(x) to reach an integer > n when started at n, or -1 if no such integer is ever reached.
In the creation of this sequence I have tested all integers from 0 to 400000000 and they all reach an integer.

Links

Crossrefs

Cf. A085068, A085071, A085328, A085330.

Programs

  • Mathematica
    f[n_] := Block[{c = 1, k = 4 n/3}, While[ !IntegerQ@k, c++; k = 4 Ceiling@k/3]; c]; t = Table[0, {1000}]; Do[ a = f@n; If[a < 101 && t[[a]] == 0, t[[a]] = n; Print[{a, n}]], {n, 0, 2800000}]

A087790 Partial sums of A085068.

Original entry on oeis.org

1, 4, 6, 7, 9, 18, 19, 27, 30, 31, 38, 40, 41, 43, 49, 50, 53, 57, 58, 63, 65, 66, 68, 71, 72, 78, 82, 83, 86, 88, 89, 91, 95, 96, 101, 104, 105, 109, 111, 112, 114, 118, 119, 122, 130, 131, 135, 137, 138, 140, 143, 144, 148, 155, 156, 159, 161, 162, 164, 171, 172, 176
Offset: 0

Views

Author

Benoit Cloitre, Oct 07 2003

Keywords

Formula

Conjecture : a(n)=3n+O(sqrt(n))

A129122 Numbers k such that A085068(3*k+1) = 7.

Original entry on oeis.org

3, 38, 60, 62, 75, 107, 119, 132, 156, 164, 188, 213, 245, 300, 332, 357, 381, 389, 413, 426, 438, 470, 483, 485, 507, 542, 564, 566, 623, 651, 708, 710, 732, 767, 789, 791, 804, 836, 848, 861, 885, 893, 917, 942, 974, 1029, 1061, 1086, 1110, 1118, 1142, 1155
Offset: 1

Views

Author

Benoit Cloitre, Aug 18 2007

Keywords

Programs

  • Mathematica
    f[n_] := Block[{c = 1, k = 4 n/3}, While[ ! IntegerQ[k], c++; k = 4*Ceiling[k]/3]; c]; Select[Range[2000], f[3*# + 1] == 7 &] (* Stefan Steinerberger, Aug 20 2007 *)

Extensions

More terms from Stefan Steinerberger, Aug 20 2007

A083514 Number of steps for iteration of map x -> (4/3)*ceiling(x) to reach an integer > 3n+1 when started at 3n+1, or -1 if no such integer is ever reached.

Original entry on oeis.org

3, 2, 8, 7, 2, 3, 5, 2, 6, 3, 2, 5, 4, 2, 3, 4, 2, 4, 3, 2, 4, 9, 2, 3, 5, 2, 6, 3, 2, 5, 5, 2, 3, 6, 2, 5, 3, 2, 7, 4, 2, 3, 4, 2, 4, 3, 2, 4, 5, 2, 3, 6, 2, 5, 3, 2, 8, 8, 2, 3, 7, 2, 7, 3, 2, 6, 4, 2, 3, 4, 2, 4, 3, 2, 4, 7, 2, 3, 8, 2, 8, 3, 2, 6, 6, 2, 3, 5, 2, 8, 3, 2, 5, 4, 2, 3, 4, 2, 4, 3, 2, 4, 6, 2, 3
Offset: 0

Views

Author

Benoit Cloitre, Sep 28 2003

Keywords

Comments

It is conjectured that an integer is always reached.
Also number of steps for iteration of map x -> (4/3)*floor(x) to reach an integer when started at 3n+4.

Links

  • J. C. Lagarias and N. J. A. Sloane, Approximate squaring (pdf, ps), Experimental Math., 13 (2004), 113-128.

Crossrefs

Equals A085068(3n+1).

Programs

  • Maple
    b:= proc(n) local i; n; for i do 4/3*ceil(%);
      if %::integer then return i fi od
    end:
a:= n-> b(3*n+1):
seq(a(n), n=0..100);  # Alois P. Heinz, Mar 01 2021
  • PARI
    a(n)=if(n<0,0,c=(3*n+1)*4/3; x=1; while(frac(c)>0,c=4/3*ceil(c); x++); x)
  • PARI
    a(n)=if(n<0,0,c=(3*n+4)*4/3; x=1; while(frac(c)>0,c=4/3*floor(c); x++); x)

Formula

a(3n+1)=2.
Showing 1-7 of 7 results.

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

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