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.
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, 241, 27190, 2, 51, 21, 11, 260, 3, 1, 62, 26, 14, 8, 151, 2, 73, 31, 17, 492, 268, 3, 1, 86, 2535, 869, 315546, 1065, 183, 2, 99, 43, 2226, 15, 350, 294, 3, 1, 114, 50, 1457, 18
Offset: 1
Keywords
Programs
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Mathematica
Table[Length[NestWhileList[# Ceiling[Sqrt[#]]/Floor[Sqrt[#]]&,n, !IntegerQ[ Sqrt[#]]&]],{n,70}] (* Harvey P. Dale, Oct 23 2016 *) -
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)
Formula
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.
Comments