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

1,2

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

Links

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

Example

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.

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

Prog

(PARI) A321992(n)={if(2<n=prime(n), forprime(q=2, , Mod(2, n)^(q-1)==1 && q!=n && return(q)), 0)}

Crossrefs

Cf. A002326, A014664, A065091.

Sequence in context: A232610 A235337 A151994 * A231806 A183782 A341751

Adjacent sequences: A321989 A321990 A321991 * A321993 A321994 A321995

Keyword

nonn,look

Author

M. F. Hasler, Mar 15 2019

Status

approved