A082991 - OEIS

%I #20 Nov 07 2017 18:31:00

%S 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,

%T 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,

%U 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

%N 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.

%C 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).

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

%C From _Antti Karttunen_, Nov 07 2017: (Start)

%C 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.

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

%C (End)

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

%H Antti Karttunen, <a href="/A082991/b082991.txt">Table of n, a(n) for n = 1..16385</a>

%F a(1) = 1; for n > 1, a(n) = 2*A095955(n). [See comments.] - _Antti Karttunen_, Nov 07 2017

%e 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.

%Y Cf. A000010, A000203, A062401, A095955.

%K nonn

%O 1,2

%A _Benoit Cloitre_, May 29 2003

%E Definition simplified by _Antti Karttunen_, Nov 07 2017