A248972 a(n) is the smallest b such that b^((p-1)/2) == -1 (mod p) where p = A080076(n) is the n-th Proth prime.

2, 2, 2, 3, 3, 5, 3, 5, 7, 3, 3, 3, 5, 3, 5, 7, 3, 5, 3, 3, 3, 5, 13, 3, 3, 3, 5, 3, 5, 7, 5, 13, 3, 3, 13, 3, 11, 5, 3, 3, 3, 11, 3, 11, 3, 3, 5, 3, 7, 3, 3, 5, 3, 5, 11, 3, 3, 5, 11, 3, 7, 5, 5, 3, 5, 3, 5, 3, 3, 3, 5, 3, 3, 3, 19, 3, 3, 3, 7, 7, 3, 3, 11, 5, 3, 3, 5, 3, 11, 5, 3, 7

Offset: 1

Author

M. F. Hasler, Oct 18 2014

Keywords

• nonn

Comments

Proth's theorem asserts that p=1+k*2^m (with odd k < 2^m) is prime if there exists b such that b^((p-1)/2) == -1 (mod n). This sequence lists the smallest b which certifies primality of A080076(n) via this relation.

For n > 3, a(n) is an odd prime. - Thomas Ordowski, Apr 23 2019

Crossrefs

Cf. A080076.

A subsequence of A020649 and of A053760.

Programs

PARI

A248972(n)=my(N=A080076[n]);for(a=0,9e9,Mod(a,N)^(N\2)==-1&&return(a))
A080076=[];forprime(p=1,99999,isproth(p)&&(A080076=concat(A080076,p))&&print1(A248972(#A080076)","))
isproth(x)={ !bittest(x--, 0) & (x>>valuation(x, 2))^2 < x }

Formula

a(n) = A020649(A080076(n)) = A053760(k), where prime(k) = A080076(n). - Thomas Ordowski, Apr 23 2019