This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A156344 #8 Jul 01 2019 02:03:00
%S A156344 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 A156344 241,27190,2,51,21,11,260,3,1,62,26,14,8,151,2,73,31,17,492,268,3,1,
%U A156344 86,2535,869,315546,1065,183,2,99,43,2226,15,350,294,3,1,114,50,1457,18
%N A156344 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 A156344 We conjecture sequence is never zero.
%F A156344 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 A156344 Table[Length[NestWhileList[# Ceiling[Sqrt[#]]/Floor[Sqrt[#]]&,n, !IntegerQ[ Sqrt[#]]&]],{n,70}] (* _Harvey P. Dale_, Oct 23 2016 *)
%o A156344 (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 A156344 Cf. A002620, A073524.
%K A156344 nonn
%O A156344 1,2
%A A156344 _Benoit Cloitre_, Feb 08 2009