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

1,1

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.

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 *)

Prog

(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

Crossrefs

Sequence in context: A074380 A215672 A085999 * A137198 A194231 A141705

Adjacent sequences: A224692 A224693 A224694 * A224696 A224697 A224698

Keyword

nonn

Author

Gary Detlefs, Apr 15 2013

Extensions

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

Status

approved