A363796 a(n) is the least prime p such that p^n + 2*n is prime, or -1 if there is no such p.

3, 3, 5, 3, 13, 31, 5, 3, 71, 3, 1279, -1, 641, 3, 11, -1, 3, 5, 317, 2477, 191, -1, 2473, -1, 59, 31, 5, -1, 3061, 907, 401, -1, 353, 3, 1153, 431, 113, 1949, 7, -1, 7027, 1063, 23, 2239, 1109, -1, 2887, 41, 251, 953, 2543, -1, 367, 607, 59, 1627, 43, -1, 67, -1, 5, 307, 257, -1, 1483, -1, 353

Offset

1,1

Comments

If n = (q^2-1)/2 + k*(q^2-q) for some odd prime q and integer k, then the only possible p is q, as otherwise q | p^n + 2*n. Conjecture: these are the only cases where a(n) = -1.

Links

Robert Israel, Table of n, a(n) for n = 1..400

Example

a(5) = 13 because 13^5 + 2*5 = 371303 is prime, and no smaller prime than 13 works.

Maple

N:= 100: # for a(1) to a(N)
V:= Vector(N):
q:=3:
do
   k0:= (q^2-1)/2;
   if k0 > N then break fi;
   for k from k0 to N by q^2-q do
     v:= q^k + 2*k; if isprime(v) then V[k]:= q;  else V[k]:= -1 fi;
   od;
   q:= nextprime(q);
od:
for k from 1 to N do
  if V[k] <> 0 then next fi;
  p:= 1:
  do
    p:= nextprime(p);
    v:= p^k + 2*k;
    if isprime(v) then V[k]:= p; break fi;
 od od:
convert(V, list);

Crossrefs

Cf. A363797, A363798.

Sequence in context: A298398 A094439 A122037 * A201454 A008316 A335952

Adjacent sequences: A363793 A363794 A363795 * A363797 A363798 A363799

Keyword

sign

Author

Robert Israel, Jun 22 2023

Status

approved