A309284 a(n) is the smallest odd composite k such that prime(n)^((k-1)/2) == -1 (mod k) and b^((k-1)/2) == 1 (mod k) for every natural b < prime(n).

3277, 5173601, 2329584217, 188985961, 5113747913401, 30990302851201, 2528509579568281, 5189206896360728641, 12155831039329417441

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 A000229 of primes.

Conjecture: the smallest quadratic non-residue modulo a(n) is prime(n), i.e., A020649(a(n)) = prime(n).

a(10) <= 41154189126635405260441. - Daniel Suteu, Jul 22 2019

Crossrefs

Cf. A000229, A001567, A006970, A307767, A309285.

Programs

PARI

isok(n,k) = (k%2==1) && !isprime(k) && Mod(prime(n), k)^((k-1)/2) == Mod(-1, k) && !for(b=2, prime(n)-1, if(Mod(b, 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, b^((a(n)-1)/2) == (b / a(n)) (mod a(n)) for every natural b <= prime(n), where (x / y) is the Jacobi symbol.

Extensions

a(5)-a(9) from Amiram Eldar, Jul 21 2019