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 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. 24
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS 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 <= 3000. The Mathematica program automatically adjusts the modulus m required to find the first integral iterate. - T. D. Noe, Apr 10 2006 LINKS T. D. Noe, Table of n, a(n) for n=1..3000 J. C. Lagarias and N. J. A. Sloane, Approximate squaring (pdf, ps), Experimental Math., 13 (2004), 113-128. EXAMPLE a(7) = 3 since 8/7 -> 16/7 -> 48/7 -> 48. MATHEMATICA 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

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Last modified December 7 18:12 EST 2019. Contains 329847 sequences. (Running on oeis4.)