A309285 a(n) is the smallest odd composite k such that prime(n)^((k-1)/2) == 1 (mod k) and q^((k-1)/2) == -1 (mod k) for every prime q < prime(n).

341, 29341, 48354810571, 493813961816587, 32398013051587

Offset: 1

Author

Thomas Ordowski, Jul 21 2019

Keywords

• nonn
• hard
• more

Comments

a(n) is an Euler pseudoprime to base 2, so it is also a Fermat pseudoprime to base 2.

This sequence is analogous to the sequence A307965 of primes.

Conjecture: the smallest prime quadratic residue modulo a(n) is prime(n).

a(6) <= 35141256146761030267, a(7) <= 4951782572086917319747. - Daniel Suteu, Jul 22 2019

Crossrefs

Cf. A001567, A006970, A307798, A307965, A309284.

Programs

PARI

isok(n,k) = (k%2==1) && !isprime(k) && Mod(prime(n), k)^((k-1)/2) == Mod(1, k) && !forprime(q=2, prime(n)-1, if(Mod(q, k)^((k-1)/2) != Mod(-1, k), return(0)));
a(n) = for(k=9, oo, if(isok(n, k), return(k))); \\ Daniel Suteu, Jul 22 2019

Formula

According to the data, for n > 1, q^((a(n)-1)/2) == (q / a(n)) (mod a(n)) for every prime q <= prime(n), where (x / y) is the Jacobi symbol.

Extensions

a(4)-a(5) from Amiram Eldar, Jul 21 2019