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.

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

Offset: 1

Author

N. J. A. Sloane, Aug 29 2002

Keywords

• nonn
• nice

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 <= 100. - Roland Bacher, Aug 30 2002

Always reaches an integer for n <= 200. - N. J. A. Sloane, Sep 04 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

Examples

			a(7) = 3 since 8/7 -> 16/7 -> 48/7 -> 48.
		

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.

Crossrefs

Cf. A073528, A073529, A068119, A001511, A072340, A075102, A073341.

Programs

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[kT. D. Noe, Apr 10 2006 *)

Extensions

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

a(19) and a(21) onwards from Roland Bacher, Aug 30 2002