A377208 a(n) is the number of iterations that n requires to reach a noninteger or a Fibonacci number under the map x -> x / z(x), where z(k) = A007895(k) is the number of terms in the Zeckendorf representation of k; a(n) = 0 if n is a Fibonacci number.

0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 2, 0, 2, 1, 1, 1, 2, 1, 1, 0, 2, 1, 3, 1, 1, 2, 1, 1, 2, 1, 1, 1, 0, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 0, 2, 1, 2, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 2, 1, 1, 3, 1, 1, 1, 2, 1, 2, 1, 2, 3, 1, 1, 2, 1, 1, 1, 1, 0, 3, 1, 2, 2, 2, 1, 2, 1, 1, 2, 1

Offset: 1

Author

Amiram Eldar, Oct 20 2024

Keywords

• nonn
• easy
• base

Comments

The Fibonacci numbers are the fixed points of the map, since z(Fibonacci(k)) = 1 for all k >= 1. Therefore they are arbitrarily assigned the value a(Fibonacci(k)) = 0.

Each number n starts a chain of a(n) integers: n, n/z(n), (n/z(n))/z(n/z(n)), ..., of them the first a(n)-1 integers are Zeckendorf-Niven numbers (A328208).

Examples

			a(12) = 2 since 12/z(12) = 4 and 4/z(4) = 2 is a Fibonacci number that is reached after 2 iterations.
a(36) = 3 since 36/z(36) = 18, 18/z(18) = 9 and 9/z(9) = 9/2 is a noninteger that is reached after 3 iterations.
		

Links

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

Crossrefs

Cf. A000005, A000045, A007895, A328208, A376615 (binary analog), A377209, A377210.

Programs

Mathematica

zeck[n_] := Length[DeleteCases[NestWhileList[# - Fibonacci[Floor[Log[Sqrt[5]*# + 3/2]/Log[GoldenRatio]]] &, n, # > 1 &], 0]]; (* Alonso del Arte at A007895 *)
a[n_] := a[n] = Module[{z = zeck[n]}, If[z == 1, 0, If[!Divisible[n, z], 1, 1 + a[n/z]]]]; Array[a, 100]

PARI

zeck(n) = if(n<4, n>0, my(k=2, s, t); while(fibonacci(k++)<=n, ); while(k && n, t=fibonacci(k); if(t<=n, n-=t; s++); k--); s) \\ Charles R Greathouse IV at A007895
a(n) = {my(z = zeck(n)); if(z == 1, 0, if(n % z, 1, 1 + a(n/z)));}

Formula

a(n) = 0 if and only if n is in A000045 (by definition).

a(n) >= 2 if and only if n is in A328208 \ A000079 (i.e., n is a Zeckendorf-Niven number that is not a Fibonacci number).

a(n) >= 3 if and only if n is in A377209 \ A000079.

a(n) >= 4 if and only if n is in A377210 \ A000079.

a(n) < A000005(n).