A082991 a(1) = 1 and for n > 1, a(n) = 2 * length of the cycle reached for the map x -> A062401(x), starting at n [where A062401(n) = phi(sigma(n))], or -1 if no finite cycle is ever reached.

1, 2, 2, 4, 2, 4, 4, 2, 2, 4, 4, 2, 4, 2, 2, 6, 4, 6, 2, 2, 6, 2, 2, 6, 6, 2, 6, 6, 2, 6, 6, 4, 6, 6, 6, 4, 6, 6, 6, 6, 2, 4, 2, 6, 6, 6, 6, 4, 4, 4, 6, 4, 6, 4, 6, 4, 4, 6, 6, 4, 6, 4, 4, 4, 6, 4, 4, 4, 4, 4, 6, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 6, 4, 4, 6, 4, 4, 6, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4

Offset: 1

Author

Benoit Cloitre, May 29 2003

Keywords

• nonn

Comments

From the original definition: Define a sequence u_n as follows: u_n(1) = n, thereafter u_n(2k) = sigma(u_n(2k-1)), u_n(2k+1) = phi(u_n(2k)); then a(n) is the length of the ultimate period of u_n(k) (which is conjectured to become ultimately periodic for any n>=1).

Conjecture: despite results for small terms, all even number are obtained as values. (For example, 12 occurs since a(12102) = 12).

From Antti Karttunen, Nov 07 2017: (Start)

Because for all n > 1, A000010(n) < n and A062401(n) > 1, such sequences u_n cannot end in odd cycle when n > 1. From this follows that for n > 1, a(n) = 2 * length of the cycle reached for the map x->A062401(x), starting at n, or -1 if no finite cycle is ever reached.

See entry A095955 for further notes about the occurrence of cycles.

(End)

Examples

			If n=6, u(1)=6, u(2)=sigma(6)=12, u(3)=phi(12)=4, u(4)=sigma(4)=7 u(5)=phi(7)=6, hence u(k) becomes periodic with period (6,12,4,7) of length 4 and a(6)=4.
		

References

• J. Berstel et al., Combinatorics on Words: Christoffel Words and Repetitions in Words, Amer. Math. Soc., 2008. See p. 83.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16385

Crossrefs

Cf. A000010, A000203, A062401, A095955.

Formula

a(1) = 1; for n > 1, a(n) = 2*A095955(n). [See comments.] - Antti Karttunen, Nov 07 2017

Extensions

Definition simplified by Antti Karttunen, Nov 07 2017