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%I #25 Nov 27 2018 03:54:49
%S 2,3,5,19,2039,2879
%N Primes prime(n) such that 1 + Sum_{k=1..n} 2^(prime(k)-1) is prime.
%C Primes prime(n) such that A080355(n+1) is prime.
%C The prime p = 19 gives the prime 332887 = 1010001010001010111_2.
%C The positions of 1's from the end are 1, 2, 3, 5, 7, 11, 13, 17, 19.
%C Let S(n) = Sum_{k=1..n} 2^(prime(k)-1). Conjecture: q(n) = 1 + S(n) is prime if and only if 2^S(n) == 1 (mod q(n)).
%e a(3) = 5 since 1 + 2^(2-1) + 2^(3-1) + 2^(5-1) = 10111_2 = 23 is prime.
%e Note that prime(3) = 5 and A080355(3+1) = 23 prime.
%t Prime@ Select[Range[10^3], PrimeQ[1 + Total@ Array[2^(Prime[#] - 1) &, #]] &] (* _Michael De Vlieger_, Oct 31 2018 *)
%o (PARI) isok(p) = isprime(p) && isprime(1 + sum(k=1, primepi(p), 2^(prime(k)-1))); \\ _Michel Marcus_, Oct 27 2018
%Y Cf. A000040, A000043, A080355, A113878.
%K nonn,more
%O 1,1
%A _Thomas Ordowski_, Oct 27 2018
%E a(5)-a(6) from _Robert Israel_, Oct 27 2018
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