A378847 -id:A378847 - cp's OEIS frontend

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

Kevin Ryde, Dec 09 2024

nonn

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).

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

Cf. A001281 (step), A135730 (number of steps).
Cf. A378846 (with halving steps), A378847 (with tripling steps).
Cf. A033491 (in 3x+1).

C
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A378846 Smallest starting x which takes n halving steps to reach the minimum of a cycle in the 3x-1 iteration.

Original entry on oeis.org

1, 2, 4, 3, 6, 11, 13, 9, 18, 35, 25, 47, 33, 63, 45, 81, 95, 117, 127, 85, 57, 113, 133, 89, 97, 65, 129, 87, 173, 225, 231, 293, 309, 377, 261, 273, 545, 671, 465, 485, 597, 647, 741, 529, 353, 705, 471, 941, 1029, 1241, 837, 577, 385, 257, 513, 343, 229, 153

Offset: 0

Kevin Ryde, Dec 15 2024

nonn

Each step is x -> 3x-1 if x odd, or x -> x/2 if x even (A001281) and here only the halving steps x/2 are counted.
The number of halving steps is A377524(x) so that a(n) = x is the smallest x for which A377524(x) = n.
a(n) <= 2*a(n-1) is an upper bound since x = 2*a(n-1) is a candidate for a(n) by first step x -> x/2.
All even terms are a(n) = 2*a(n-1), since any smaller even a(n) would imply a smaller a(n-1) by first step x -> x/2.
No term is of the form y = 6*k + 2, apart from a(1)=2, since odd x = 2*k+1 takes a tripling step to 3*x-1 = y and x is a smaller start with the same number of halvings as y.

Kevin Ryde, Table of n, a(n) for n = 0..932
Kevin Ryde, C Code (set count type HALF)

Cf. A001281 (step), A377524 (number of halving steps).

Cf. A378845 (with all steps), A378847 (with tripling steps).

C
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cp's OEIS Frontend

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

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