OFFSET
1,1
COMMENTS
The definition's congruence is verified if n is a safe prime A005385 with m the corresponding Sophie Germain prime A005384; and for a few other n, which form the sequence.
If that congruence is verified and m is prime, then n is prime (follows from a result by Fedor Petrov).
That congruence is equivalent to the combination: 2^m == +-1 (mod n) and 2^m == 2 (mod m).
LINKS
Francois R. Grieu, Table of n, a(n) for n = 1..2796 (terms <2^42).
Peter Košinár, Report of a composite n, Math StackExchange, Mar 06 2018.
Fedor Petrov, Proof that the congruence and m prime imply n prime, MathOverflow.
EXAMPLE
n = 683 = 2*341+1 is in the sequence because 2^341 == 2048 == 3*n-1 (mod 341*683) and m = 341 = 11*13 is composite.
n = 301703 = 2*150851+1 is in the sequence because 2^150851 == 301704 == n+1 (mod 150851*301703) and m = 150851 = 251*601 is composite.
n = 5 = 2*2+1 is not in the sequence because m = 2 is prime.
MATHEMATICA
For[m=1, (n=2m+1)<4444444, ++m, If[MemberQ[{n+1, 3n-1}, PowerMod[2, m, m*n]] &&!PrimeQ[m], Print[n]]] (* Francois R. Grieu, Mar 19 2018 *)
PROG
(PARI) isok(n) = {if ((n % 2) && (m=(n-1)/2) && !isprime(m), v = lift(Mod(2, m*n)^m); if ((v == n+1) || (v == 3*n-1), return (1)); ); return (0); } \\ Michel Marcus, Mar 06 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Francois R. Grieu, Mar 05 2018
STATUS
approved