A334685 Start with n, and successively apply phi, phi, sigma, phi, phi, sigma, phi, ... until reaching 1; a(n) is the number of steps needed (phi = A000010, sigma = A000203); or a(n) = -1 if 1 is never reached.

0, 1, 2, 2, 5, 2, 5, 5, 5, 5, 8, 5, 8, 5, 8, 8, 11, 5, 8, 8, 8, 8, 8, 8, 11, 8, 8, 8, 11, 8, 11, 11, 11, 11, 11, 8, 11, 8, 11, 11, 14, 8, 11, 11, 11, 8, 11, 11, 11, 11, 14, 11, 14, 8, 14, 11, 11, 11, 14, 11, 14, 11, 11, 14, 14, 11, 11, 14, 11, 11, 14, 11, 14, 11, 14, 11, 14, 11, 14, 14, 14, 14, 14, 11, 14, 11, 14, 14

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=11 is 11, 10, 4, 7, 6, 2, 3, 2, 1, 1, 1, ..., which takes eight steps to reach 1, so a(11) = 8.

Mathematica

Array[-1 + Length@ NestWhile[Append[#1, If[#2 == 0, DivisorSigma[1, #1[[-1]]], EulerPhi@ #1[[-1]] ]] & @@ {#, Mod[Length@ #, 3]} &, {#}, Last[#] > 1 &] &, 80] (* Michael De Vlieger, May 09 2020 *)

Prog

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

Crossrefs

Cf. A000010, A000203, A032452, A334686, A334523, A334725.

Sequence in context: A171889 A171868 A292146 * A340694 A101910 A162784

Adjacent sequences: A334682 A334683 A334684 * A334686 A334687 A334688

Keyword

nonn

Author

N. J. A. Sloane, May 08 2020

Status

approved