A309180 Unsuspected numbers to check in the Collatz conjecture.

61, 91, 205, 253, 325, 415, 433, 577, 637, 739, 901, 919, 991, 1063, 1171, 1225, 1333, 1387, 1549, 1663, 1711, 1837, 1873, 1891, 2035, 2125, 2197, 2287, 2359, 2449, 2521, 2683, 2791, 2845, 3007, 3169, 3187, 3277, 3331, 3349, 3439, 3493

Offset

1,1

Comments

The sequence is constructed using the following steps:

Start at 1, and color it blue. Go through the Collatz algorithm, highlight each number that is not in 'blue' in 'red' until you reach an already 'red' number or lower number that is 'blue'. Color the next uncolored number 'blue' and repeat.

So starting at 1, 1 becomes blue, then 4 becomes red, 2 becomes red and move to next number. Next uncolored number is 3, so 3 becomes blue. Then 10 becomes red, 5 becomes red, 16 red, 8 red, and 4 is already red so done. Next uncolored number is 6, so 6 becomes blue, etc.

For any number k the expected colors are:

red if k (mod 18) is equal to 2, 4, 5, 8, 10, 11, 13, 14, 16, or 17.

blue if k (mod 18) is equal to 0, 1, 3, 6, 7, 9, 12, or 15

The list here are the numbers that do not fit this pattern.

Observation:

For up to at least 180000 only numbers of the format k (mod 18) = 1 and k (mod 18) = 7 were not fitting the pattern, they were all red instead of blue.

Links

Table of n, a(n) for n=1..42.

P. Stikker, C# code to visualize grid.

P. Stikker, Excel VBA code to visualize grid.

Prog

(C#) // Unsuspected numbers to check in Collatz conjecture
using System;
namespace Collatz {
    class Program {
        static void Main() {
            Console.Write("Enter until which number to check:");
            int nMax = int.Parse(Console.ReadLine());
            int[] values = new int[nMax + 1], colors = new int[nMax + 1];
            for (int i = 1; i < nMax + 1; i++) {
                values[i] = i; colors[i] = 0;
            }
            for (int i = 1; i < nMax + 1; i++) {
                if (colors[i] == 0) {
                    var myNum = i;
                    do {
                        myNum = (myNum % 2 == 0 ? myNum / 2 : myNum * 3 + 1);
                        if (myNum > i) {
                            if (myNum <= nMax) colors[myNum] = 1;
                        }
                        else myNum = 0;
                    } while (myNum != 0);
                }
            }
            for (int i = 1; i < nMax+1; i++) {
                if (i % 18 == 0 || i % 18 == 1 || i % 18 == 3  || i % 18 == 6  ||
                    i % 18 == 7 || i % 18 == 9 || i % 18 == 12 || i % 18 == 15) {
                    if (colors[i]==1) Console.WriteLine(i);
                }
                else {
                    if (colors[i] == 0) Console.WriteLine(i);
                }
            }
            Console.ReadKey();
        }
    }
}
(PARI) isokb(k) = (k==0) || (k==1) || (k==3) || (k==6) || (k==7) || (k==9) || (k==12) || (k==15);
isokr(k) = (k==2) || (k==4) || (k==5) || (k==8) || (k==10) || (k==11) || (k==13) || (k==14) || (k==16) || (k==17);
f(n) = if(n%2, 3*n+1, n/2);
nocolor(n, vred, vblue) = !vecsearch(vred, n) && !vecsearch(vblue, n);
chk(nn) = {vblue = []; vred = []; for (n=1, nn, if (nocolor(n, vred, vblue), ok = 1; vblue = vecsort(concat(vblue, n), , 8); ntodo = n; while (1, m = f(ntodo); if (vecsearch(vred, m), break); if ((m<n) && vecsearch(vblue, m), break); if (!vecsearch(vblue, m), vred = vecsort(concat(vred, m), , 8)); ntodo = m; ); ); ); vb = select(x->(!isokb(x%18)), vblue); vr = select(x->(!isokr(x%18)), vred); select(x->x<=nn, vecsort(concat(vr, vb))); } \\ Michel Marcus, Jul 17 2019

Crossrefs

Cf. A014682 (the Collatz function). So far the numbers are all of the form 6n + 1, so this would be a subset of A016921.

Sequence in context: A103812 A047272 A123207 * A038856 A144970 A045562

Adjacent sequences: A309177 A309178 A309179 * A309181 A309182 A309183

Keyword

nonn

Author

Peter Stikker, Jul 15 2019

Status

approved