A309771 a(n) is the number of odd numbers found in the Collatz trajectory of 6*n + 3 before reaching a number x == 5 (mod 8).

2, 5, 4, 1, 18, 4, 5, 1, 2, 5, 6, 1, 3, 2, 3, 1, 2, 5, 12, 1, 5, 6, 4, 1, 2, 3, 10, 1, 5, 2, 3, 1, 2, 4, 4, 1, 5, 10, 7, 1, 2, 5, 8, 1, 3, 2, 3, 1, 2, 6, 8, 1, 4, 3, 13, 1, 2, 3, 5, 1, 5, 2, 3, 1, 2, 6, 4, 1, 6, 10, 6, 1, 2, 4, 9, 1, 3, 2, 3, 1, 2

Offset

0,1

Comments

The number reached at the end of the computation, 8x + 5, will need more than 2 even steps to get to an odd number in the Collatz trajectory.

a(3 + 4*n) = 1 always because 6*(3 + 4*n) + 3 = 21 + 24*n = 8*(3*n + 2) + 5.

There's an interesting pattern in these seemingly chaotic numbers:

a(12 + 32*n) = 3,

a(13 + 32*n) = 2,

a(14 + 32*n) = 3,

a(15 + 32*n) = a(3 + 4(8*n + 3)) = 1,

a(16 + 32*n) = 2.

A general formula for a(n), if it exists, could shed some light on the Collatz problem. The big spike at n=4 (large sequence for 27) seems a mystery for me. For big values of n we don't usually get big a(n), probably because of the presence of patterns like the above (just an observation).

Links

Máximo Pérez López, Table of n, a(n) for n = 0..20000

Wikipedia, Collatz Conjecture

Formula

Not known for now, a(n) is calculated from computer program and definition of the Collatz function, e.g., f(x) = 3x + 1 if x is odd, or x/2 if x is even.

Example

For n = 0, 6*0 + 3 = 3. Collatz sequence of 3 omitting the even numbers: 3, 5. 5 = 8*0 + 5 so the computation stops: 2 odd numbers found.
For n = 1, 6*1 + 3 = 9. Sequence: 9, 7, 11, 17, 13. 13 = 8*1 + 5, halt. a(1) = 5.

Mathematica

Array[Count[NestWhileList[If[EvenQ@ #, #/2, 3 # + 1] &, 6 # + 3, Mod[#, 8] != 5 &], _?OddQ] &, 81, 0] (* Michael De Vlieger, Sep 27 2019 *)

Prog

(C)
#include <stdio.h>
#include <stdlib.h>
int collatzOdd(int n) {
        int i = 1;
        int r;
        while ( (r = n % 8) != 5) {
                i++;
                if (r == 1) {
                        n = 6*(n / 8) + 1;
                } else {
                        n = 6*(n / 4) + 5;
                }
        }
        return i;
}
int main(int argc, char *argv[]) {
        int i = 0;
        while (i <= 20000) {
                printf("%d %d\n", i, collatzOdd((6 * i) + 3));
                i++;
        }
        return (EXIT_SUCCESS);
}

Crossrefs

Sequence in context: A011417 A231890 A087561 * A009738 A268647 A177067

Adjacent sequences: A309768 A309769 A309770 * A309772 A309773 A309774

Keyword

easy,nonn

Author

Máximo Pérez López, Aug 16 2019

Status

approved