%I A073524
%S A073524 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 A073524 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 A073524 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 A073524 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 A073524 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 A073524 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 (noe(AT)sspectra.com), Apr 10 2006
%H A073524 T. D. Noe, <a href="b073524.txt">Table of n, a(n) for n=1..3000</a>
%H A073524 J. C. Lagarias and N. J. A. Sloane, Approximate squaring (<a href="http:/
               /www.research.att.com/~njas/doc/apsq.pdf">pdf</a>, <a href="http:/
               /www.research.att.com/~njas/doc/apsq.ps">ps</a>), Experimental Math., 
               13 (2004), 113-128.
%e A073524 a(7) = 3 since 8/7 -> 16/7 -> 48/7 -> 48.
%t A073524 Table[{n, First[Flatten[Position[Map[Denominator, NestList[ # Ceiling[ 
               # ] &, (n + 1)/n, 20]], 1]]]}, {n, 1, 20}]
%t A073524 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 A073524 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 (noe(AT)sspectra.com), Apr 10 2006
%Y A073524 Cf. A073528, A073529, A068119, A001511, A072340, A075102, A073341.
%Y A073524 Sequence in context: A042903 A132534 A118702 this_sequence A130226 A032808 
               A037317
%Y A073524 Adjacent sequences: A073521 A073522 A073523 this_sequence A073525 A073526 
               A073527
%K A073524 nonn,nice
%O A073524 1,3
%A A073524 N. J. A. Sloane (njas(AT)research.att.com), Aug 29 2002
%E A073524 a(5) - a(10), a(12) - a(18), a(20) = 11 from Ed Pegg jr, Aug 29, 2002
%E A073524 T. D. Noe also found a(5) and remarks that the final integer is 9.5329600...*10^57734. 
               - Aug 29 2002
%E A073524 a(11) from T. D. Noe, who remarks that the final integer is 5.131986636061311...*10^13941166 
               - Aug 29 2002
%E A073524 a(19) and a(21) onwards from Roland Bacher, Aug 30, 2002
%E A073524 Always reaches an integer for n <= 100. - Roland Bacher, Aug 30, 2002
%E A073524 Always reaches an integer for n <= 200. - N. J. A. Sloane (njas(AT)research.att.com), 
               Sep 04, 2002
%E A073524 Always reaches an integer for n <= 500 by comparing results with index 
               1000 and index 2500 - Robert G. Wilson v, Sep 11, 2002