A005602 Smallest prime beginning a complete Cunningham chain of length n (of the first kind).

13, 3, 41, 509, 2, 89, 1122659, 19099919, 85864769, 26089808579, 665043081119, 554688278429, 4090932431513069, 95405042230542329, 90616211958465842219, 810433818265726529159

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

Comments

The word "complete" indicates each chain is exactly n primes long (i.e., the chain cannot be a subchain of another one). Except for a(1), each term, by definition, is a Sophie Germain prime (A005384) as is each element except the last in each chain; each element after the first in each chain is a safe prime (A005385), so interior elements are both.

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• C. Caldwell, The Prime Number Glossary: Cunningham Chains.
• G. Löh, Long chains of nearly doubled primes, Math. Comp. vol. 53 no. 188 (1989) pp 751-759.
• W. Roonguthai, Yves Gallot's Proth.exe and Cunningham Chains
• Jonathan P. Sorenson, Jonathan Webster, Two Algorithms to Find Primes in Patterns, arXiv:1807.08777 [math.NT], 2018.

Crossrefs

Cf. (A005384 and A005385), A007700, A023272, A023302, A023330, A057331, A005603.

Extensions

a(13) found by Jack Brennen; a(14) found by Paul Jobling (Paul.Jobling(AT)WhiteCross.com) [Oct 23 2000]

Better description from Rick L. Shepherd, Jul 07 2004

a(15) found by Jonathan Webster and Jonathan Sorenson, added Jun 26 2018

a(16) found by Phil Carmody and Paul Jobling, Feb 2002, and added by Mauro Fiorentini, Feb 21 2025