A085071 -id:A085071 - cp's OEIS frontend

A085068 Number of steps >= 1 for iteration of map x -> (4/3)*ceiling(x) to reach an integer when started at n, or -1 if no such integer is ever reached.

1, 3, 2, 1, 2, 9, 1, 8, 3, 1, 7, 2, 1, 2, 6, 1, 3, 4, 1, 5, 2, 1, 2, 3, 1, 6, 4, 1, 3, 2, 1, 2, 4, 1, 5, 3, 1, 4, 2, 1, 2, 4, 1, 3, 8, 1, 4, 2, 1, 2, 3, 1, 4, 7, 1, 3, 2, 1, 2, 7, 1, 4, 3, 1, 9, 2, 1, 2, 6, 1, 3, 6, 1, 5, 2, 1, 2, 3, 1, 6, 5, 1, 3, 2, 1, 2, 8, 1, 5, 3, 1, 5, 2, 1, 2, 5, 1, 3, 4, 1, 6

Offset: 0

N. J. A. Sloane, Aug 11 2003

It is conjectured that an integer is always reached.

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

Cf. A085058, A085071, A085328, A085330, A083514.

f := x->(4/3)*ceil(x); g := proc(n) local t1,c; global f; t1 := f(n); c := 1; while not type(t1, 'integer') do c := c+1; t1 := f(t1); od; RETURN([c,t1]); end;
# second Maple program:
a:= proc(n) local i; n; for i do 4/3*ceil(%);
      if %::integer then return i fi od
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Mar 01 2021

f[n_] := Block[{c = 1, k = 4 n/3}, While[ ! IntegerQ@k, c++; k = 4 Ceiling@k/3]; c]; Table[f@n, {n, 0, 104}] (* Robert G. Wilson v *)

from fractions import Fraction
def A085068(n):
    c, x, m = 1, Fraction(4*n,3), Fraction(4,3)
    while x.denominator > 1:
        x = m*x._ceil_()
        c += 1
    return c # 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

N. J. A. Sloane, Aug 13 2003

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

Cf. A085068, A085071, A085330.

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 *)

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

N. J. A. Sloane, Aug 13 2003

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

Cf. A085068, A085071, A085328.

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 *)

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)","))) }

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

A087675 Consider recurrence b(0) = (2n+1)/2, b(n) = b(0)*floor(b(n-1)); sequence gives first integer reached.

Original entry on oeis.org

5, 35, 18, 814, 39, 390, 68, 72827, 105, 1449, 150, 31887, 203, 3596, 264, 27852510, 333, 7215, 410, 208464, 495, 12690, 588, 10561998, 689, 20405, 798, 744049, 915, 30744, 1040, 46620858503, 1173, 44091, 1314, 1950450, 1463, 60830, 1620, 121575329, 1785

Offset: 2

N. J. A. Sloane, following a suggestion of Bela Bajnok (bbajnok(AT)gettysburg.edu), Sep 27 2003

Robert Israel, Table of n, a(n) for n = 2..10000
J. C. Lagarias and N. J. A. Sloane, Approximate squaring (pdf, ps), Experimental Math., 13 (2004), 113-128.

A001511 gives number of steps to reach an integer.

Cf. A001511, A081853, A085071, A057016.

f:= proc(n)
  local b0, b;
  b0:= (2*n+1)/2;
  b:= b0;
  do
    b:= b0*floor(b);
    if b::integer then return b fi
  od
end proc:
map(f, [$2..100]); # Robert Israel, Nov 25 2019

f[n_] := Module[{b0, b}, b0 = (2n+1)/2; b = b0; While[True, b = b0*Floor[b]; If[IntegerQ[b], Return[b]]]];
Table[f[n], {n, 2, 100}] (* Jean-François Alcover, Oct 23 2023, after Robert Israel *)

The even-indexed terms are given by A007742.

Offset corrected by Robert Israel, Nov 25 2019

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

Robert G. Wilson v, May 27 2007

nonn

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.

Robert G. Wilson v, Table of n, a(n) for n = 1..50.

Cf. A085068, A085071, A085328, A085330.

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}]

cp's OEIS Frontend

A085068 Number of steps >= 1 for iteration of map x -> (4/3)*ceiling(x) to reach an integer when started at n, or -1 if no such integer is ever reached.

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A085328 Record values in A085068.

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A085330 Where records occur in A085068.

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A087675 Consider recurrence b(0) = (2n+1)/2, b(n) = b(0)*floor(b(n-1)); sequence gives first integer reached.

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A129377 First occurrence of n in A085068.

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