A378845 Smallest starting x which takes n steps to reach the minimum of a cycle in the 3x-1 iteration.

1, 2, 4, 7, 3, 6, 11, 19, 21, 13, 26, 9, 18, 35, 37, 73, 25, 49, 98, 33, 66, 131, 45, 90, 175, 127, 117, 85, 149, 57, 113, 199, 209, 133, 265, 89, 177, 65, 119, 237, 87, 159, 165, 329, 231, 225, 439, 309, 293, 585, 377, 391, 273, 261, 521, 1042, 671, 695, 485

Offset

0,2

Comments

Each step is x -> 3x-1 if x odd, or x -> x/2 if x even (A001281).

The number of steps is A135730(x) so that a(n) = x is the smallest x for which A135730(x) = n.

a(n) <= 2*a(n-1) since x = 2*a(n-1) is a candidate for a(n) by first step x -> x/2.

Even terms are always a(n) = 2*a(n-1) since any smaller even a(n) would imply a smaller a(n-1) after first step x -> x/2.

No term is of the form 12*k+4, since its first step to 6*k+2 is also where the first step from 2*k+1 goes and the latter is a smaller start.

a(n) >= (a(n-1) + 1)/3 is a lower bound since a(n) = x must at least have a first step 3x-1 which reaches somewhere with n-1 further steps, so 3x-1 >= a(n-1).

Equality a(n) = (a(n-1) + 1)/3 = x occurs iff that x is an odd integer and not a cycle minimum, so its first step is to 3x-1 = a(n-1) (as for example at n=11).

Links

Kevin Ryde, Table of n, a(n) for n = 0..1435

Kevin Ryde, C Code

Prog

(C) /* See links. */

Crossrefs

Cf. A001281 (step), A135730 (number of steps).

Cf. A378846 (with halving steps), A378847 (with tripling steps).

Cf. A033491 (in 3x+1).

Sequence in context: A256998 A356432 A303641 * A137282 A139696 A308049

Adjacent sequences: A378842 A378843 A378844 * A378846 A378847 A378848

Keyword

nonn

Author

Kevin Ryde, Dec 09 2024

Status

approved