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

Author

Kevin Ge, Oct 28 2024

Keywords

• nonn

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.

Examples

			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.
		

Crossrefs

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

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

Programs

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