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A096822 Smallest primes of form p = 2^x-(2n-1) where x=A096502(n), the least exponent providing this kind of prime.

%I A096822 #19 Jul 06 2021 01:49:40
%S A096822 3,5,3,549755813881,7,5,3,17,47,13,11,41,7,5,3,97,31,29,2011,89,23,
%T A096822 536870869,19,17,79,13,11,73,7,5,3,193,191,61,59,953,439,53,179,433,
%U A096822 47,173,43,41,167,37,163,929,31,29,67108763,409,23,149,19,17,911,13,11,137
%N A096822 Smallest primes of form p = 2^x-(2n-1) where x=A096502(n), the least exponent providing this kind of prime.
%C A096822 If 2n-1 is a provable Riesel number (A101036), then there exists a finite set of primes P(2n-1) such that every 2^x-(2n-1) > 0 is divisible by p(x) in P(2n-1). If some 2^x-(2n-1) = p(x), then a(n) = p(x). Otherwise, p(x) is a proper divisor of 2^x-(2n-1), which must be composite, and no a(n) exists.
%C A096822 For example, if n = 254602, then 2n-1 = 509203 is a provable Riesel number. Every 2^x-509203 > 0 is divisible by prime p(x) in P(509203) = {3,5,7,13,17,241}.  2^x-509203 > 0 implies x >= 19 implies 2^x-509203 > 241 >= p(x), so p(x) is a proper divisor and every 2^x-509203 is composite. Hence a(254602) does not exist.
%H A096822 T. D. Noe, <a href="/A096822/b096822.txt">Table of n, a(n) for n = 1..935</a>
%e A096822 a(1) = 3 is the first Mersenne prime;
%e A096822 a(64) = 2^47 - 127 = 140737488355201, where 47 = A096502(64), 127 = 2*64 - 1.
%t A096822 f[n_]:=Module[{lst={},exp=Ceiling[Log[2,1+n]]},While[!PrimeQ[2^exp-n],exp++]; AppendTo[lst,2^exp-n]]; Flatten[f/@Range[1,1001,2]] (* _Ivan N. Ianakiev_, Mar 08 2016 *)
%Y A096822 Cf. A096502.
%K A096822 nonn
%O A096822 1,1
%A A096822 _Labos Elemer_, Jul 13 2004