A109927 - OEIS

%I #10 Apr 03 2023 10:36:10

%S 3,5,11,23,37,83,157,179,359,661,719,877,997,1019,1237,1439,1657,2039,

%T 2063,2137,2459,2557,2819,2903,2963,3023,3061,3623,3779,3803,3863,

%U 4177,4261,4357,4621,4919,5399,5581,5639,6037,6121,6217,6361,6899,6983,7079

%N First primes p connected to two primes either by 2p+1 or 2p-1 upward [downward (p-1)/2 or (p+1)/2].

%C These primes may be part of Cunningham chains longer than three terms. It seems the two operators are never mixed, except for 3, 5 and 7: -for 3, we have: 2 through <2p-1> -> 3 through <2p+1> -> 7 -for 5: 3 <2p-1> -> 5 <2p+1> -> 11 -for 7: 3 <2p+1> -> 7 <2p-1> -> 13

%C For p > 7, such a mixed chain with p in the middle is impossible because the number 3 would be a nontrivial factor of either the smallest or the largest term. - _Jeppe Stig Nielsen_, May 05 2019

%C Primes (excluding 2 and 7) that divide more than one semiprime triangular number A068443. - _Jeppe Stig Nielsen_, May 05 2019

%C The disjoint union of A059455 and A109835. - _Jeppe Stig Nielsen_, May 05 2019

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

%e a(3)=11 is here because 5->11->23 through <2p+1>;

%e a(4)=23 because 11->23->47 through <2p+1>;

%e a(5)=37 because 19->37->73 through <2p-1>.

%o Terms computed by Gilles Sadowski.

%o (PARI) forprime(p=3,10^6,if(p%3==2,isprime((p-1)/2)&&isprime(2*p+1),isprime((p+1)/2)&&isprime(2*p-1))&&print1(p,", ")) \\ _Jeppe Stig Nielsen_, May 05 2019

%Y Cf. A005382, A005383, A005384, A005385, A059455, A068497.

%K easy,nonn

%O 1,1

%A _Alexandre Wajnberg_, Aug 31 2005