OFFSET
1,1
COMMENTS
Equivalently, numbers n = 2m+1 that are not safe primes A005385 even though n and m are strong probable primes (that is, prime or strong pseudoprime A001262). That follows from a result by Fedor Petrov.
All known terms are prime, including the 542622 less than 2^65 (obtained by post-processing Jan Feitsma and William Galway's table).
LINKS
Jan Feitsma, The pseudoprimes below 2^64
William Galway, Tables of pseudoprimes and related data [Includes a file with pseudoprimes up to 2^64.]
Fedor Petrov, The question's congruence and m prime imply n prime, MathOverflow.
EXAMPLE
n = 715523 is in the sequence because n = 2m+1 with m = 357761, and 2^m mod n = 715522 which is among 1 or n-1 (the latter), and m is a strong pseudoprime A001262. The latter holds because m = 131*2731 is composite, and m passes the strong probable prime test. The latter holds because when writing m-1 as d*(2^s) with d odd, it holds that 2^d mod m = 1 or there exists an r with 0 <= r < s and 2^(d*(2^r)) == -1 (mod m); specifically, d = 2795, s = 7, 2^2795 mod 357761 = 357760 = m-1, thus 2^(d*(2^r)) == -1 (mod m) for r = 0.
MATHEMATICA
For[m=3, (n=2m+1)<13^8, m+=2, If[MemberQ[{1, n-1}, PowerMod[2, m, n]]&&(d=m-1; t=1; While[EvenQ[d], d/=2; ++t]; If[(x=PowerMod[2, d, m])!=1, While[--t>0&&x!=m-1, x=Mod[x^2, m]]]; t>0)&&!PrimeQ[m], Print[n]]]
PROG
(PARI) is_A001262(n, a=2)={ (bittest(n, 0) && !isprime(n) && n>8) || return; my(s=valuation(n-1, 2)); if(1==a=Mod(a, n)^(n>>s), return(1)); while(a!=-1 && s--, a=a^2); a==-1; } \\ after A001262
isok(n) = if (n%2, my(m = (n-1)/2, r = Mod(2, n)^m); ((r==1) || (r==-1)) && is_A001262(m)); \\ derived from Michel Marcus, May 07 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Francois R. Grieu, Mar 25 2018
STATUS
approved