A321992 a(n) is the least prime q different from p = prime(n) such that 2^(q-1) == 1 (mod p), or 0 if no such prime exists.

0, 5, 13, 13, 31, 37, 41, 37, 67, 113, 11, 73, 61, 29, 139, 157, 233, 181, 199, 211, 19, 157, 739, 23, 193, 401, 307, 743, 37, 29, 29, 521, 409, 277, 593, 31, 53, 487, 499, 1033, 1069, 541, 571, 97, 1373, 397, 421, 149, 1583, 457, 59, 953, 73, 101, 17, 787, 1609, 541, 461

Offset: 1

Author

M. F. Hasler, Mar 15 2019

Keywords

• nonn
• look

Comments

a(n) = 0 only for n = 1, p = 2. For any odd prime, a prime q meeting the requirement does exist.

a(n) is the smallest prime q <> p such that q == 1 (mod ord_{p}(2)), where ord_{p}(2) = A002326((p-1)/2) = A014664(n). Strong conjecture: a(n) < A014664(n)^2. - Thomas Ordowski, Mar 15 2019

Examples

			For n = 2, p = prime(2) = 3, the least prime q different from 3 such that 2^(q-1) == 1 (mod 3) is a(2) = 5.
For n = 3, p = prime(3) = 5, the least prime q different from 5 such that 2^(q-1) == 1 (mod 5) is a(3) = 13.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000 (a(1) = 0 inserted by Georg Fischer, May 05 2019)

Crossrefs

Cf. A002326, A014664, A065091.

Programs

Maple

f:= proc(n) local p,q,v,j;
  if n = 1 then return 0 fi;
  p:= ithprime(n);
  v:= numtheory:-order(2,p);
  for q from 1 by v do
    if q <> p and isprime(q) then return q fi
  od
end proc:
map(f, [$1..100]); # Robert Israel, Mar 17 2019

PARI

A321992(n)={if(2