OFFSET
1,1
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.
John Armitage found Cunningham chains of length 19 and 20. It is possible a(19) = 27692774713034525184617669 and a(20) = 75949683268775452969189499. He claims the chain of length 20 is the smallest possible. - Jonathan Pappas, Aug 26 2026
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
John Armitage, Problems in Parameterization, Zero-Estimates, Point-Counting, and O-Minimality, 2021.
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 and Jonathan Webster, Two Algorithms to Find Primes in Patterns, arXiv:1807.08777 [math.NT], 2018-2019.
CROSSREFS
KEYWORD
nonn,more,changed
AUTHOR
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
a(17)-a(18) added by Jonathan Pappas, Aug 22 2026
STATUS
approved