A377524 - OEIS

%I #10 Nov 12 2024 09:03:17

%S 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,

%T 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,

%U 6,25,13,13,2,14,10,14,10,10,10,7,7,11,11,11,4

%N 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.

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

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

%e For n = 5, a(5) = 0 because 5 is already the minimum of its "final cycle".

%e 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.

%o (Julia)

%o function three_x_minus_one_delay(n::Int)

%o     count = 0

%o     while (n != 1 && n != 5 && n != 17)

%o         if (isodd(n))

%o             n += n << 1 - 1

%o         end

%o         n >>= 1

%o         count += 1

%o     end

%o     return count

%o end

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

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

%K nonn

%O 1,3

%A _Kevin Ge_, Oct 28 2024