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

%I #4 Oct 20 2024 13:55:51

%S 1,4,12,24,180,1056,2592,15552,46656,544320,20528640,238085568,

%T 3547348992,46438023168,599501979648

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

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

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

%e n | The n iterations

%e --+---------------------------------------------

%e 1 | 4 -> 2 = Fibonacci(3)

%e 2 | 12 -> 4 -> 2

%e 3 | 24 -> 12 -> 4 -> 2

%e 4 | 180 -> 60 -> 30 -> 10 -> 5 = Fibonacci(5)

%e 5 | 1056 -> 264 -> 66 -> 22 -> 11 -> 11/2

%e 6 | 2592 -> 1296 -> 324 -> 108 -> 27 -> 9 -> 9/2

%t zeck[n_] := Length[DeleteCases[NestWhileList[# - Fibonacci[Floor[Log[Sqrt[5]*# + 3/2]/Log[GoldenRatio]]] &, n, # > 1 &], 0]]; (* _Alonso del Arte_ at A007895 *)

%t s[n_] := s[n] = Module[{z = zeck[n]}, If[z == 1, 0, If[!Divisible[n, z], 1, 1 + s[n/z]]]];

%t 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 (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 s(n) = {my(z = zeck(n)); if(z == 1, 0, if(n % z, 1, 1 + s(n/z)));}

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

%Y Subsequence of A328208.

%K nonn,base,more

%O 0,2

%A _Amiram Eldar_, Oct 20 2024