Smallest Cunningham Chain of the First Kind of Length 6

Theorem
Let $p$ be a Sophie Germain prime.

Let the sequence $S_p$ be defined for $n \ge 1$ as:
 * $n_k = \begin{cases} p & : k = 1 \\

2 n_{k - 1} + 1 & : k > 1 \end{cases}$

The smallest $p$ for which the first $6$ terms of $S_p$ are all primes is $89$:
 * $89, 179, 359, 719, 1439, 2879$

The last in the sequence is, by definition, not a Sophie Germain prime.

Proof
By definition, a Sophie Germain prime is a prime number such that $2 p + 1$ is also prime.

Let $P: \Z \to \Z$ be the mapping defined as:
 * $P \left({n}\right) = 2 n + 1$

Applying $P$ iteratively to each of the smallest Sophie Germain primes in turn:

Thus the length of $S_2$ is $5$, and that of $S_5, S_{11}, S_{23}$ is $4, 3, 2$ respectively.

Hence the sequence of $6$:
 * $89, 179, 359, 719, 1439, 2879$