A265416 - OEIS

%I #14 Mar 19 2024 17:23:23

%S 0,1,1,2,6,1,5,3,5,7,9,2,4,6,6,4,59,6,61,8,63,10,14,3,12,5,5,7,7,7,58,

%T 5,10,60,11,7,10,62,13,9,13,64,42,11,11,15,66,4,55,13,13,6,68,6,57,8,

%U 15,8,10,8,48,59,10,6,10,10,14,61,72,12,12,8,12,10

%N Number of steps needed to reach 1 or to enter the cycle in the "sqrt(Pi)*x+1" problem.

%C The sqrt(Pi)*x+1 problem is as follows: start with a number x. If x is even, divide it by 2, otherwise multiply it by sqrt(Pi) and add 1, and then take the integer part.

%C There are three possible behaviors for such trajectories when n>0:

%C (i) The trajectory reaches 1 (and enters the "trivial" cycle 2-1-2-1-2...).

%C (ii) Cyclic trajectory. The trajectory becomes periodic and the period does not contain a 1.

%C (iii) The trajectory is divergent (I conjecture that this cannot occur).

%C For many numbers, the element of the trivial cycle is 1, except for the numbers: 3, 6, 7, 12, 13, 14, 15, 23, 24, 26, 27, 28, 29, 30, 33, 35, 37, 39, 41, 46, 48, 52, 54, 56, 58, 59, 60, 63, ...

%H Michel Lagneau, <a href="/A265416/b265416.txt">Table of n, a(n) for n = 1..10000</a>

%e a(5) = 6 because 5 -> 9 -> 16 -> 8 -> 4 -> 2 -> 1 with 6 iterations where:

%e 5 -> floor(5*sqrt(Pi)+1) = 9;

%e 9 -> floor(9*sqrt(Pi)+1) = 16;

%e 16 -> 16/2 = 8 -> 8/2 = 4 -> 4/2 = 2 -> 2/2 = 1, the end of the cycle.

%e a(6) = 1 because 6 -> 3 with 1 iteration where:

%e 6 -> 3;

%e 3 -> floor(3*sqrt(Pi)+1) = 6, the end of the cycle.

%p A265416:= proc(n)

%p     local cyc, x;

%p     x := n;

%p     cyc := {x} ;

%p     for s from 0 do

%p         if {1} intersect cyc = {1} then

%p             return s;

%p         end if;

%p         if type(x, 'even') then

%p             x := x/2 ;

%p         else

%p             x := floor(evalf(sqrt(Pi))*x+1) ;

%p         end if;

%p         if {x} intersect cyc = {x} and s > 0 then

%p             return s;

%p         end if;

%p         cyc := cyc union {x} ;

%p     end do:

%p end proc: # Program from R.J. Mathar adapted for this sequence - see A264789.

%t Table[Length@ NestWhileList[If[EvenQ@ #, #/2, Floor[# Sqrt@ Pi + 1]] &, n, UnsameQ, All] - 2, {n, 0, 72}] (* Program from Michael De Vlieger adapted for this sequence - see A264789 *).

%Y Cf. A006577, A264789.

%K nonn

%O 1,4

%A _Michel Lagneau_, Dec 08 2015