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 $q_{i+1}=2q_{i}+1$ 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 $k$.