A265416 Number of steps needed to reach 1 or to enter the cycle in the "sqrt(Pi)*x+1" problem.

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

Offset

1,4

Comments

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.

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

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

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

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

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, ...

Links

Michel Lagneau, Table of n, a(n) for n = 1..10000

Example

a(5) = 6 because 5 -> 9 -> 16 -> 8 -> 4 -> 2 -> 1 with 6 iterations where:
5 -> floor(5*sqrt(Pi)+1) = 9;
9 -> floor(9*sqrt(Pi)+1) = 16;
16 -> 16/2 = 8 -> 8/2 = 4 -> 4/2 = 2 -> 2/2 = 1, the end of the cycle.
a(6) = 1 because 6 -> 3 with 1 iteration where:
6 -> 3;
3 -> floor(3*sqrt(Pi)+1) = 6, the end of the cycle.

Maple

A265416:= proc(n)
    local cyc, x;
    x := n;
    cyc := {x} ;
    for s from 0 do
        if {1} intersect cyc = {1} then
            return s;
        end if;
        if type(x, 'even') then
            x := x/2 ;
        else
            x := floor(evalf(sqrt(Pi))*x+1) ;
        end if;
        if {x} intersect cyc = {x} and s > 0 then
            return s;
        end if;
        cyc := cyc union {x} ;
    end do:
end proc: # Program from R.J. Mathar adapted for this sequence - see A264789.

Mathematica

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 *).

Crossrefs

Cf. A006577, A264789.

Sequence in context: A220959 A360598 A243434 * A199953 A364682 A076039

Adjacent sequences: A265413 A265414 A265415 * A265417 A265418 A265419

Keyword

nonn

Author

Michel Lagneau, Dec 08 2015

Status

approved