Cunningham chain of the first kind

Definition

Let $k$ be a natural number. A Cunningham chain of the first kind of length $k$ is a sequence of primes $q_{1} < q_{2} < \dots < q_{k}$ such that $q_{i + 1} = 2 q_{i} + 1$ for each $1 \leq i \leq k - 1$ .

A complete Cunningham chain of the first kind is a Cunningham chain of the first kind that cannot be extended further in either direction.

Given a Cunningham chain of the first kind of length $2$ , the first prime in the chain is a Sophie Germain prime and the second prime in the chain is a safe prime. More generally, in any Cunningham chain of length $k$ , the first $k - 1$ primes are Sophie Germain primes and the last $k - 1$ primes are safe primes.

Related facts and conjectures

Conjecture on existence of Cunningham chains of the first kind of arbitrary length

Relation with other properties

Cunningham chain of the second kind
Bitwin chain is a combination of Cunningham chains of both kinds and twin primes.

Testing/listing

The ID of the sequence in the Online Encyclopedia of Integer Sequences is A005602

This lists, for every $k$ , the smallest prime beginning a complete Cunningham chain of length $k$ .