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

%I A377211 #10 Apr 26 2025 06:01:28
%S A377211 1,4,12,24,180,1056,2592,15552,46656,544320,20528640,238085568,
%T A377211 3547348992,46438023168,599501979648
%N A377211 a(n) is the least number k such that A377208(k) = n, or -1 if no such number exists.
%C A377211 a(15) > 2.4*10^12, if it exists.
%C A377211 All the terms are Zeckendorf-Niven numbers (A328208).
%e A377211   n | The n iterations
%e A377211   --+---------------------------------------------
%e A377211   1 | 4 -> 2 = Fibonacci(3)
%e A377211   2 | 12 -> 4 -> 2
%e A377211   3 | 24 -> 12 -> 4 -> 2
%e A377211   4 | 180 -> 60 -> 30 -> 10 -> 5 = Fibonacci(5)
%e A377211   5 | 1056 -> 264 -> 66 -> 22 -> 11 -> 11/2
%e A377211   6 | 2592 -> 1296 -> 324 -> 108 -> 27 -> 9 -> 9/2
%t A377211 zeck[n_] := Length[DeleteCases[NestWhileList[# - Fibonacci[Floor[Log[Sqrt[5]*# + 3/2]/Log[GoldenRatio]]] &, n, # > 1 &], 0]]; (* _Alonso del Arte_ at A007895 *)
%t A377211 s[n_] := s[n] = Module[{z = zeck[n]}, If[z == 1, 0, If[!Divisible[n, z], 1, 1 + s[n/z]]]];
%t A377211 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]
%o A377211 (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
%o A377211 s(n) = {my(z = zeck(n)); if(z == 1, 0, if(n % z, 1, 1 + s(n/z)));}
%o A377211 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; }
%Y A377211 Cf. A000045, A376619 (binary analog), A377208.
%Y A377211 Subsequence of A328208.
%K A377211 nonn,base,more
%O A377211 0,2
%A A377211 _Amiram Eldar_, Oct 20 2024