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

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

Author

Kevin Ryde, Dec 15 2024

Keywords

• nonn

Comments

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.

Links

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

Crossrefs

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

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

Programs

C

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