A059761 - OEIS

%I #27 Feb 16 2025 08:32:44

%S 3,29,53,113,131,173,191,233,239,251,281,293,419,431,443,491,593,641,

%T 653,659,683,743,761,809,911,953,1013,1049,1103,1223,1289,1499,1559,

%U 1583,1601,1733,1973,2003,2069,2129,2141,2273,2339,2351,2393,2399,2543

%N Initial primes of Cunningham chains of first type with length exactly 2. Primes in A059453 that survive as primes only one "2p-1 iteration", forming chains of exactly 2 terms.

%C Primes p such that {(p-1)/2, p, 2p+1, 4p+3} = {composite, prime, prime, composite}.

%H Amiram Eldar, <a href="/A059761/b059761.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Harvey P. Dale)

%H Chris Caldwell's Prime Glossary, <a href="https://t5k.org/glossary/page.php?sort=CunninghamChain">Cunningham chains</a>.

%H Warut Roonguthai, <a href="http://web.archive.org/web/20010405230842/http://ksc9.th.com/warut/cunningham.html">Yves Gallot's Proth.exe and Cunningham Chains</a>. [Wayback Machine link]

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CunninghamChain.html">Cunningham Chain</a>.

%e 53 is a term because 26 and 215 are composites, and 53 and 107 are primes.

%t ccftQ[p_]:=Boole[PrimeQ[{(p-1)/2,p,2 p+1,4 p+3}]]=={0,1,1,0}; Select[ Prime[ Range[400]],ccftQ] (* _Harvey P. Dale_, Jun 19 2021 *)

%Y Cf. A023272, A023302, A023330, A005384, A005385, A059452, A059453, A059454, A059455, A007700.

%K nonn

%O 1,1

%A _Labos Elemer_, Feb 20 2001