A377524 Number of steps for n to reach the minimum of its final cycle under iterations of the map (A123684): x->(3x-1)/2 if x odd, x/2 otherwise; or -1 if this never happens.

0, 1, 3, 2, 0, 4, 2, 3, 7, 1, 5, 5, 6, 3, 7, 4, 0, 8, 5, 2, 5, 6, 2, 6, 10, 7, 4, 4, 8, 8, 4, 5, 12, 1, 9, 9, 9, 6, 10, 3, 6, 6, 7, 7, 14, 3, 11, 7, 11, 11, 8, 8, 12, 5, 8, 5, 20, 9, 9, 9, 5, 5, 13, 6, 25, 13, 13, 2, 14, 10, 14, 10, 10, 10, 7, 7, 11, 11, 11, 4

Offset

1,3

Comments

The currently known cycle minimums are 1, 5, 17 and there are no known a(n) = -1 (trajectory never reaches a cycle).

This sequence is one way to extend A006666 (number of Collatz (3x+1)/2 steps) to the negative numbers.

Links

Table of n, a(n) for n=1..80.

Example

For n = 5, a(5) = 0 because 5 is already the minimum of its "final cycle".
For n = 12, a(12) = 6 because 12 takes 6 iterations to reach the minimum of its "final cycle": 12 -> 6 -> 3 -> 8 -> 4 -> 2 -> 1.

Prog

(Julia)
function three_x_minus_one_delay(n::Int)
    count = 0
    while (n != 1 && n != 5 && n != 17)
        if (isodd(n))
            n += n << 1 - 1
        end
        n >>= 1
        count += 1
    end
    return count
end

Crossrefs

Cf. A123684 ((3x-1)/2 map), A135730 (all steps).

Cf. A006666 (for (3x+1)/2).

Sequence in context: A290794 A328178 A127571 * A194662 A143612 A353160

Adjacent sequences: A377521 A377522 A377523 * A377525 A377526 A377527

Keyword

nonn

Author

Kevin Ge, Oct 28 2024

Status

approved