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%I #29 Nov 20 2020 09:10:50
%S 0,1,2,3,18,2,3,4,6,7,26,4,9,3,4,8,6,4,56,11,3,4,42,4,33,7,5,4,38,5,
%T 79,6,4,15,14,8,200,29,13,5,36,3,4,5,7,10,11,8,6,20,47,27,43,9,41,9,
%U 10,23,37,17,18,6,7,6,32,15,225,7,73,11,20,12,182,9,16,7,10,15,196,8
%N Number of steps to reach an integer starting with (n+1)/n and using the map x -> x*ceiling(x); or -1 if no integer is ever reached.
%C Computed by doing all computations over the integers (multiply by n) and by truncating modulo n^250. This avoids the explosion of the integers (of order 2^(2^k) after k iterations) and gives the correct answer if the final index i(n) is < 250 (or perhaps 249 or 248). If the algorithm does not stop before 245 one should increase precision (work with n^500 or even higher). - Roland Bacher
%C Always reaches an integer for n <= 100. - _Roland Bacher_, Aug 30 2002
%C Always reaches an integer for n <= 200. - _N. J. A. Sloane_, Sep 04 2002
%C Always reaches an integer for n <= 500 by comparing results with index 1000 and index 2500. - _Robert G. Wilson v_, Sep 11 2002
%C Always reaches an integer for n <= 3000. The Mathematica program automatically adjusts the modulus m required to find the first integral iterate. - _T. D. Noe_, Apr 10 2006
%C Always reaches an integer for n <= 5000. - _Ben Branman_, Feb 12 2011
%H T. D. Noe, <a href="/A073524/b073524.txt">Table of n, a(n) for n=1..3000</a>
%H J. C. Lagarias and N. J. A. Sloane, Approximate squaring (<a href="http://neilsloane.com/doc/apsq.pdf">pdf</a>, <a href="http://neilsloane.com/doc/apsq.ps">ps</a>), Experimental Math., 13 (2004), 113-128.
%e a(7) = 3 since 8/7 -> 16/7 -> 48/7 -> 48.
%t Table[{n, First[Flatten[Position[Map[Denominator, NestList[ # Ceiling[ # ] &, (n + 1)/n, 20]], 1]]]}, {n, 1, 20}]
%t f[n_] := Block[{k = (n + 1)/n, c = 0}, While[ !IntegerQ[k], c++; k = Mod[k*Ceiling[k], n^250]]; c]; Table[ f[n], {n, 1, 100}]
%t Table[lim=50; While[k=0; x=1+1/n; m=n^lim; While[k<lim-3 && !IntegerQ[x], x=Mod[x*Ceiling[x],m]; k++ ]; k==lim-3, lim=2*lim]; k, {n,1000}] (* _T. D. Noe_, Apr 10 2006 *)
%Y Cf. A073528, A073529, A068119, A001511, A072340, A075102, A073341.
%K nonn,nice
%O 1,3
%A _N. J. A. Sloane_, Aug 29 2002
%E a(5)-a(10), a(12)-a(18), a(20) = 11 from _Ed Pegg Jr_, Aug 29 2002
%E _T. D. Noe_ also found a(5) and remarks that the final integer is 9.5329600...*10^57734. - Aug 29 2002
%E a(11) from _T. D. Noe_, who remarks that the final integer is 5.131986636061311...*10^13941166 - Aug 29 2002
%E a(19) and a(21) onwards from _Roland Bacher_, Aug 30 2002
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