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%I #8 Jul 01 2019 02:03:00
%S 1,2,3,1,6,2,9,3,1,14,103,2,19,7,3,1,26,10,105,2,33,13,312,3,1,42,691,
%T 241,27190,2,51,21,11,260,3,1,62,26,14,8,151,2,73,31,17,492,268,3,1,
%U 86,2535,869,315546,1065,183,2,99,43,2226,15,350,294,3,1,114,50,1457,18
%N Number of steps to reach a square starting from n and iterating the map: x -> x*ceiling(sqrt(x))/floor(sqrt(x)) or zero if no square is reached.
%C We conjecture sequence is never zero.
%F a(k^2)=1, a(k*(k+1))=2, a(k*(k+2))=3, and less trivially it appears a(floor(n^2/4)+1) = 1 + ceiling((n-1)^2/2) and then the square reached is (floor(n^2/4)+1)^2.
%t Table[Length[NestWhileList[# Ceiling[Sqrt[#]]/Floor[Sqrt[#]]&,n, !IntegerQ[ Sqrt[#]]&]],{n,70}] (* _Harvey P. Dale_, Oct 23 2016 *)
%o (PARI) a(n)=if(n<0,0,s=n;c=1;while(frac(sqrt(s))>0, s=s*ceil(sqrt(s))/floor(sqrt(s)); c++);c)
%Y Cf. A002620, A073524.
%K nonn
%O 1,2
%A _Benoit Cloitre_, Feb 08 2009
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