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%I #14 Jul 29 2013 03:48:35
%S 3,3,1,3,2,2,1,7,4,4,1,2,2,4,1,6,3,5,1,5,2,2,1,4,6,3,1,2,2,3,1,7,3,4,
%T 1,3,2,2,1,7,4,7,1,2,2,5,1,3,3,10,1,4,2,2,1,3,5,11,1,2,2,3,1,5,3,3,1,
%U 3,2,2,1,4,4,6,1,2,2,4,1,4,3,6,1,6,2,2,1,6,7,3,1,2,2,3,1,4,3,5,1,3,2,2,1,4
%N Number of steps to reach an integer starting with n + 1/4 and iterating the map x -> x*ceiling(x).
%C If the initial value is n + 1/2 we get A001511.
%C S(n)=sum(k=1, n, a(k)) seems to be asymptotic to 3*n. S(n)=3n for in A074069.
%C The sign of 3n-S(n) seems to change often: 3n-S(n) gives A074077. Is 3n-S(n) bounded? - _Benoit Cloitre_, Sep 05 2002
%H Reinhard Zumkeller, <a href="/A068119/b068119.txt">Table of n, a(n) for n = 1..10000</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.
%F a(n) = 1 if n == 3 (mod 4); a(n) = 2 if n == 5, 6, 12, 13 (mod 16); a(n) = 3 if n == 1, 2, 4, 17, 26, 30, 33, 36, 48, 49, 56, 62 (mod 64);...
%t ce[n_] := Length[NestWhileList[#*Ceiling[#] &, n + 1/4, ! IntegerQ[#] &]] - 1; ce /@ Range[104] (* _Jayanta Basu_, Jul 29 2013 *)
%o (PARI) a(n)=if(n<0,0,s=n+1/4; c=0; while(frac(s)>0,s=s*ceil(s); c++); c)
%o (Haskell)
%o import Data.Ratio ((%), denominator)
%o a068119 n = fst $ until ((== 1) . denominator . snd)
%o (\(i, x) -> (i + 1, f x)) (0, fromInteger n + 1%4)
%o where f x = x * fromIntegral (ceiling x)
%o -- _Reinhard Zumkeller_, May 26 2013
%Y Cf. A073524, A074069, A074077
%K nonn
%O 1,1
%A _Benoit Cloitre_, Aug 30 2002
%E Corrected by Diego Torres (torresvillarroel(AT)hotmail.com), Aug 31 2002
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