A005602 - OEIS

%I M4885 #38 Nov 05 2025 15:35:27

%S 13,3,41,509,2,89,1122659,19099919,85864769,26089808579,665043081119,

%T 554688278429,4090932431513069,95405042230542329,90616211958465842219,

%U 810433818265726529159

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

%C 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.

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

%H C. Caldwell, <a href="https://t5k.org/glossary/page.php?sort=CunninghamChain">The Prime Number Glossary: Cunningham Chains</a>.

%H G. Löh, <a href="https://www.jstor.org/stable/2008735">Long chains of nearly doubled primes</a>, Math. Comp. vol. 53 no. 188 (1989) pp 751-759.

%H W. Roonguthai, <a href="http://ksc9.th.com/warut/cunningham.html">Yves Gallot's Proth.exe and Cunningham Chains</a>

%H Jonathan P. Sorenson, Jonathan Webster, <a href="https://arxiv.org/abs/1807.08777">Two Algorithms to Find Primes in Patterns</a>, arXiv:1807.08777 [math.NT], 2018.

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

%K nonn

%O 1,1

%A _N. J. A. Sloane_

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

%E Better description from _Rick L. Shepherd_, Jul 07 2004

%E a(15) found by _Jonathan Webster_ and Jonathan Sorenson, added Jun 26 2018

%E a(16) found by _Phil Carmody_ and Paul Jobling, Feb 2002, and added by _Mauro Fiorentini_, Feb 21 2025