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

1,2

Comments

We conjecture sequence is never zero.

Links

Table of n, a(n) for n=1..68.

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.

Mathematica

Table[Length[NestWhileList[# Ceiling[Sqrt[#]]/Floor[Sqrt[#]]&, n, !IntegerQ[ Sqrt[#]]&]], {n, 70}] (* Harvey P. Dale, Oct 23 2016 *)

Prog

(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)

Crossrefs

Cf. A002620, A073524.

Sequence in context: A083855 A062565 A175137 * A218796 A119440 A165742

Adjacent sequences: A156341 A156342 A156343 * A156345 A156346 A156347

Keyword

nonn

Author

Benoit Cloitre, Feb 08 2009

Status

approved