A377211 a(n) is the least number k such that A377208(k) = n, or -1 if no such number exists.

1, 4, 12, 24, 180, 1056, 2592, 15552, 46656, 544320, 20528640, 238085568, 3547348992, 46438023168, 599501979648

Offset: 0

Author

Amiram Eldar, Oct 20 2024

Keywords

• nonn
• base
• more

Comments

a(15) > 2.4*10^12, if it exists.

All the terms are Zeckendorf-Niven numbers (A328208).

Examples

			  n | The n iterations
  --+---------------------------------------------
  1 | 4 -> 2 = Fibonacci(3)
  2 | 12 -> 4 -> 2
  3 | 24 -> 12 -> 4 -> 2
  4 | 180 -> 60 -> 30 -> 10 -> 5 = Fibonacci(5)
  5 | 1056 -> 264 -> 66 -> 22 -> 11 -> 11/2
  6 | 2592 -> 1296 -> 324 -> 108 -> 27 -> 9 -> 9/2
		

Crossrefs

Cf. A000045, A376619 (binary analog), A377208.

Subsequence of A328208.

Programs

Mathematica

zeck[n_] := Length[DeleteCases[NestWhileList[# - Fibonacci[Floor[Log[Sqrt[5]*# + 3/2]/Log[GoldenRatio]]] &, n, # > 1 &], 0]]; (* Alonso del Arte at A007895 *)
s[n_] := s[n] = Module[{z = zeck[n]}, If[z == 1, 0, If[!Divisible[n, z], 1, 1 + s[n/z]]]];
seq[len_] := Module[{v = Table[0, {len}], c = 0, k = 1, i}, While[c < len, i = s[k] + 1; If[v[[i]] == 0, c++; v[[i]] = k]; k++]; v]; seq[9]

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
s(n) = {my(z = zeck(n)); if(z == 1, 0, if(n % z, 1, 1 + s(n/z)));}
lista(len) = {my(v = vector(len), c = 0, k = 1, i); while(c < len, i = s(k) + 1; if(v[i] == 0, c++; v[i] = k); k++); v; }