A224695 Pseudoprimes to a twin prime criterion of Aebi and Cairns.

561, 645, 1905, 2465, 4369, 4371, 23001, 25761, 60701, 87249, 158369, 181901, 206601, 212421, 253241, 280601, 332949, 348161, 513629, 580337, 587861, 656601, 831405, 1207361, 1275681, 1472505, 1489665, 1678541, 1735841, 1815465, 1909001, 2113665, 2290641, 2419385

Offset: 1

Author

Gary Detlefs, Apr 15 2013

Keywords

• nonn

Comments

Theorem 3 in the paper (see link) states that if p and p+2 are both prime then 2^(p+2) == 3*p+8 mod p(p+2). This sequence lists the pseudoprimes to this criterion.

Of the first 27 terms listed, 16 are divisible by 3.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..903 (terms below 10^11)
• Christian Aebi and Grant Cairns, Catalan numbers, primes and twin primes, Elemente der Mathematik, Vol. 63, No. 4 (2008), pp. 153-164.

Programs

Maple

for n from 1 to 1500000 do if 2^(n+2) mod n*(n+2)=3*n+8 and not isprime(n) then print(n) fi od;

Mathematica

aQ[n_] := CompositeQ[n] && PowerMod[2, n + 2,  n*(n + 2)] == 3*n + 8; Select[
Range[10^6], aQ] (* Amiram Eldar, Jul 12 2019 *)

PARI

isok(n) = (n>1) && !isprime(n) && Mod(2, n*(n+2))^(n+2) == Mod(3*n+8, n*(n+2)); \\ Michel Marcus, Jul 12 2019

Extensions

a(14) and more terms from Amiram Eldar, Jul 12 2019