A334686 Start with n, and successively apply phi, phi, sigma', phi, phi, sigma', phi, ... until reaching either 0 or 1; a(n) is the number of steps needed (phi = A000010, sigma' = A001065); or a(n) = -1 if 0 or 1 is never reached.

0, 1, 2, 2, 3, 2, 3, 3, 3, 3, 5, 3, 5, 3, 5, 5, 6, 3, 5, 5, 5, 5, 6, 5, 6, 5, 5, 5, 8, 5, 6, 6, 6, 6, 6, 5, 8, 5, 6, 6, 8, 5, 8, 6, 6, 6, 6, 6, 8, 6, 8, 6, 8, 5, 8, 6, 8, 8, 8, 6, 8, 6, 8, 8, 8, 6, 8, 8, 8, 6, 8, 6, 8, 8, 8, 8, 8, 6, 8, 8, 8, 8, 9, 6, 9, 8, 8, 8, 9, 6, 8, 8, 8, 6, 8, 8, 9, 8, 8, 8, 9, 8, 9, 8, 8, 8

Offset

1,3

Comments

Created following a suggestion from R. J. Mathar in an attempt to understand A032452.

Links

Rémy Sigrist, Table of n, a(n) for n = 1..25000

L. Alaoglu and P. Erdős, A conjecture in elementary number theory, Bull. Amer. Math. Soc. 50 (1944), 881-882.

Example

The trajectory of n=7 is 7, 6, 2, 1, ...  which takes three steps to reach 0 or 1, so a(7) = 3.

Mathematica

A334686[n_]:=Module[{i=0}, NestWhile[If[Mod[i++, 3]==2, DivisorSigma[1, #]-#, EulerPhi[#]]&, n, #>1&]; i]; Array[A334686, 100] (* Paolo Xausa, Nov 16 2023 *)

Prog

(PARI) a(n) = { for (k=0, oo, if (n<=1, return (k), k%3==2, n=sigma(n)-n, n=eulerphi(n))) } \\ Rémy Sigrist, May 09 2020

Crossrefs

Cf. A000010, A000203, A001065, A032452, A334685, A334523, A334725.

Sequence in context: A076223 A076235 A268058 * A057941 A358550 A126071

Adjacent sequences: A334683 A334684 A334685 * A334687 A334688 A334689

Keyword

nonn

Author

N. J. A. Sloane, May 08 2020

Status

approved