  Let $p$ be a prime number. A chain $\{p,2p+1,4p+3,\cdots,(p+1)2^{l(p)-1}-1\}$
is called the Cunningham chain generated by $p$ if all elements are prime
number and $(p+1)2^{l(p)}-1$ is composite. Then $l(p)$ is called the length of
the Cunningham chain. It is conjectured by Bateman and Horn in 1962 that the
number of prime $p\leq N$ such that $l(p)\geq k$ is asymptotically equal to
$B_k N/(\log N)^k$ with a real $B_k>0$ for all natural number $k$. This
suggests that $l(p)=\Omega(\log p/\log\log p)$. However, so far no good
estimation is known. It has not even been proven whether $\limsup_{p\to\infty}
l(p)$ is infinite or not. All we know is that $l(p)=5$ if $p=2$ and $l(p)<p$
for odd $p$ by Fermat's little theorem. Let $\alpha\geq3$ be an integer. In
this article, a generalized Fibonacci sequence
$\mathcal{F}_\alpha=\{F_n\}_{n=0}^\infty$ is defined as $F_0=0,F_1=1,
F_{n+2}=\alpha F_{n+1}+F_n (n\geq0)$, and
${}_{\mathcal{F}_\alpha}\sigma(n)=\sum_{d\mid n, 0<d\in\mathcal{F}_\alpha}d$ is
called a divisor function on $\mathcal{F}_\alpha$. Then we obtain an
interesting relation between the iteration of ${}_{\mathcal{F}_\alpha}\sigma$
and the length of Cunningham chains. For two primes $p$ and $q$, the fact
$p=2q+1$ or $2q-1$ is equivalent to
${}_{\mathcal{F}_\alpha}\sigma({}_{\mathcal{F}_\alpha}\sigma(F_p))={}_{\mathcal{F}_\alpha}\sigma(F_q)$
for some $\alpha$. By this relation, we get $l(p)\ll\log p$ under a certain
condition. It seems that this sufficient condition is plausible by numerical
test. Furthermore, the condition, written in terms of prime numbers, can be
replaced by the condition written in terms of natural numbers. This implies
that the problem of upper estimation of $l(p)$ is reduced to that on natural
numbers.
