A364800 The number of iterations that n requires to reach 1 under the map x -> A356874(x).

0, 1, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5

Offset: 1

Author

Amiram Eldar, Aug 08 2023

Keywords

• nonn
• base
• easy

Comments

a(n) is well-defined since A356874(1) = 1, and A356874(n) < n for n >= 2.

Examples

			For n = 3 the trajectory is 3 -> 2 -> 1. The number of iterations is 2, thus a(3) = 2.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A356874.

Similar sequences: A003434, A364801.

Programs

Mathematica

f[n_] := f[n] = Module[{d = IntegerDigits[n, 2], nd}, nd = Length[d]; Total[d * Fibonacci[Range[nd, 1, -1]]]]; (* A356874 *)
a[n_] := Length@ NestWhileList[f, n, # > 1 &] - 1; Array[a, 100]

PARI

f(n) = {my(b = binary(n), nb = #b); sum(i = 1, nb, b[i] * fibonacci(nb - i + 1));} \\ A356874
a(n) = if(n == 1, 0, a(f(n)) + 1);

Formula

a(n) = a(A356874(n)) + 1, for n >= 2.