A Cunningham chain of the first kind is a sequence of prime numbers (p1,...,pn) such that for all 1 ≤ i < n, pi+1 = 2 pi + 1. (Hence each term of such a chain except the last one is a Sophie Germain prime, and each term except the first is a safe prime). Similarly, a Cunningham chain of the second kind is a sequence of prime numbers (p1,...,pn) such that for all 1 ≤ i < n, pi+1 = 2 pi - 1.
Cunningham chains are also sometimes generalized to sequences of prime numbers (p1,...,pn) such that for all 1 ≤ i < n, pi+1 = api + b for fixed coprimeintegersa, b; the resulting chains are called generalized Cunningham chains.
A Cunningham chain is called complete if it cannot be further extended, i.e., if the next term in the chain would not be a prime number anymore.
Sequence A005602 (http://www.research.att.com/projects/OEIS?Anum=A005602) in OEIS: the first term of the lowest complete Cunnigham Chains of the first kind of length n, for 1 <= n <= 14
Sequence A005603 (http://www.research.att.com/projects/OEIS?Anum=A005603) in OEIS: the first term of the lowest complete Cunnigham Chains of the second kind with length n, for 1 <= n <= 15
The first, Cunninghamchains of the first kind, are sequences of primes where each is twice the preceeding prime plus one (so the first prime in such a chain is a Sophie Germain prime).
Cunninghamchains of the second kind are each twice the preceeding prime minus one.
Chains of length n of nearly doubled primes from Sloan's sequences - The least prime which generates a Cunninghamchain (second kind) of length n.
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