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

1,12

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).

Links

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

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).

Example

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.

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]

Prog

(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))); }

Crossrefs

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

Sequence in context: A111330 A225152 A117447 * A328775 A053250 A364259

Adjacent sequences: A377205 A377206 A377207 * A377209 A377210 A377211

Keyword

nonn,easy,base

Author

Amiram Eldar, Oct 20 2024

Status

approved