# Cunningham chain of the first kind

## Definition

Let  be a natural number. A Cunningham chain of the first kind of length  is a sequence of primes  such that  for each .

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 , 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 , the first  primes are Sophie Germain primes and the last  primes are safe primes.

## Testing/listing

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

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