A109927 First primes p connected to two primes either by 2p+1 or 2p-1 upward [downward (p-1)/2 or (p+1)/2].

3, 5, 11, 23, 37, 83, 157, 179, 359, 661, 719, 877, 997, 1019, 1237, 1439, 1657, 2039, 2063, 2137, 2459, 2557, 2819, 2903, 2963, 3023, 3061, 3623, 3779, 3803, 3863, 4177, 4261, 4357, 4621, 4919, 5399, 5581, 5639, 6037, 6121, 6217, 6361, 6899, 6983, 7079

Offset: 1

Author

Alexandre Wajnberg, Aug 31 2005

Keywords

• easy
• nonn

Comments

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

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

Primes (excluding 2 and 7) that divide more than one semiprime triangular number A068443. - Jeppe Stig Nielsen, May 05 2019

The disjoint union of A059455 and A109835. - Jeppe Stig Nielsen, May 05 2019

Examples

			a(3)=11 is here because 5->11->23 through <2p+1>;
a(4)=23 because 11->23->47 through <2p+1>;
a(5)=37 because 19->37->73 through <2p-1>.
		

Links

• Chris Caldwell's Prime Glossary, Cunningham chains.

Crossrefs

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

Programs

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