A377208 -id:A377208 - cp's OEIS frontend

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

Amiram Eldar, Oct 20 2024

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

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

			  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

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

Subsequence of A328208.

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]

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; }

A377384 a(n) is the number of iterations that n requires to reach a noninteger or a factorial number under the map x -> x / f(x), where f(k) = A034968(k) is the sum of digits in the factorial-base representation of k; a(n) = 0 if n is a factorial number.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Oct 27 2024

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

Each number n starts a chain of a(n) integers: n, n/f(n), (n/f(n))/f(n/f(n)), ..., of them the first a(n)-1 integers are factorial-base Niven numbers (A118363).

			a(8) = 2 since 8/f(8) = 4 and 4/f(4) = 2 is a factorial number that is reached after 2 iterations.
a(27) = 3 since 27/f(27) = 9, 9/f(9) = 3 and 3/f(3) = 3/2 is a noninteger that is reached after 3 iterations.

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

Cf. A000005, A000142, A034968, A118363, A286607, A377385, A377386.

Analogous sequences: A376615 (binary), A377208 (Zeckendorf).

fdigsum[n_] := Module[{k = n, m = 2, r, s = 0}, While[{k, r} = QuotientRemainder[k, m]; k != 0 || r != 0, s += r; m++]; s]; a[n_] := a[n] = Module[{s = fdigsum[n]}, If[s == 1, 0, If[!Divisible[n, s], 1, 1 + a[n/s]]]]; Array[a, 100]

fdigsum(n) = {my(k = n, m = 2, r, s = 0); while([k, r] = divrem(k, m); k != 0 || r != 0, s += r; m++); s;}
a(n) = {my(f = fdigsum(n)); if(f == 1, 0, if(n % f, 1, 1 + a(n/f)));}

def f(n, p=2): return n if nMichael S. Branicky, Oct 27 2024

a(n) = 0 if and only if n is in A000142 (by definition).
a(n) = 1 if and only if n is in A286607.
a(n) >= 2 if and only if n is in A118363 \ A000142 (i.e., n is a factorial-base Niven number that is not a factorial number).
a(n) >= 3 if and only if n is in A377385 \ A000142.
a(n) >= 4 if and only if n is in A377386 \ A000142.
a(n) < A000005(n).

cp's OEIS Frontend

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

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