A350969 Let phi^(k) denote the k-th iterate of phi (A000010); a(n) is smallest positive k such that phi^(k)(Fibonacci(n)) = 1.

1, 1, 1, 2, 3, 3, 4, 4, 5, 6, 7, 6, 8, 8, 8, 9, 9, 10, 11, 12, 11, 13, 13, 13, 15, 16, 15, 16, 17, 17, 19, 18, 19, 19, 20, 20, 21, 22, 22, 23, 26, 23, 25, 25, 26, 27, 28, 27, 28, 30, 28, 31, 32, 30, 34, 32, 33, 34, 35, 34, 38, 37, 36, 37, 39, 38, 40, 39, 40, 40

Offset

1,4

Comments

a(n) <= n.

The Fibonacci Quarterly asks what the range of a(n) is. For example, is a(n) ever equal to 14 or 24?

Links

Alois P. Heinz, Table of n, a(n) for n = 1..450

Douglas Lind (Proposer), Problem B-51, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 2, No. 3 (1964), p. 232; Solution, ibid., Vol. 60, No. 1 (2022), pp. 83-84.

S. Sivasankaranarayana Pillai, On a function connected with phi(n), Bull. Amer. Math. Soc., Vol. 35, No. 6 (1929), pp. 837-841.

Formula

a(n) = A049108(A000045(n)) - 1, for n > 2. - Amiram Eldar, Mar 03 2022.

a(n) = A003434(A000045(n)) for n > 2. - Alois P. Heinz, Mar 03 2022

Example

Iterating phi, F_7 = 13 -> 12 -> 4 -> 2 -> 1 takes 4 steps to reach 1, so a(7) = 4.

Maple

a:= proc(n) uses numtheory; local f, k;
      f:= phi((<<0|1>, <1|1>>^n)[1, 2]);
      for k while f>1 do f:= phi(f) od; k
    end:
seq(a(n), n=1..70);  # Alois P. Heinz, Mar 03 2022

Mathematica

a[1] = a[2] = 1; a[n_] := Length@NestWhileList[EulerPhi, Fibonacci[n], # > 1 &] - 1; Array[a, 100] (* Amiram Eldar, Mar 03 2022 *)

Crossrefs

Cf. A000010, A000045, A003434, A049108, A060607.

Sequence in context: A309077 A057365 A014245 * A096386 A257063 A273663

Adjacent sequences: A350966 A350967 A350968 * A350970 A350971 A350972

Keyword

nonn

Author

N. J. A. Sloane, Mar 03 2022

Status

approved