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A073524
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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.
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24
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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, 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, 10, 23, 37, 17, 18, 6, 7, 6, 32, 15, 225, 7, 73, 11, 20, 12, 182, 9, 16, 7, 10, 15, 196, 8
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OFFSET
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1,3
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COMMENTS
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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
Always reaches an integer for n <= 100. - Roland Bacher, Aug 30 2002
Always reaches an integer for n <= 500 by comparing results with index 1000 and index 2500. - Robert G. Wilson v, Sep 11 2002
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
Always reaches an integer for n <= 5000. - Ben Branman, Feb 12 2011
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LINKS
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J. C. Lagarias and N. J. A. Sloane, Approximate squaring (pdf, ps), Experimental Math., 13 (2004), 113-128.
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EXAMPLE
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a(7) = 3 since 8/7 -> 16/7 -> 48/7 -> 48.
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MATHEMATICA
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Table[{n, First[Flatten[Position[Map[Denominator, NestList[ # Ceiling[ # ] &, (n + 1)/n, 20]], 1]]]}, {n, 1, 20}]
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}]
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 *)
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CROSSREFS
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KEYWORD
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nonn,nice
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AUTHOR
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EXTENSIONS
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a(5)-a(10), a(12)-a(18), a(20) = 11 from Ed Pegg Jr, Aug 29 2002
T. D. Noe also found a(5) and remarks that the final integer is 9.5329600...*10^57734. - Aug 29 2002
a(11) from T. D. Noe, who remarks that the final integer is 5.131986636061311...*10^13941166 - Aug 29 2002
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STATUS
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approved
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