A321084 Primes prime(n) such that 1 + Sum_{k=1..n} 2^(prime(k)-1) is prime.

2, 3, 5, 19, 2039, 2879

Offset: 1

Author

Thomas Ordowski, Oct 27 2018

Keywords

• nonn
• more

Comments

Primes prime(n) such that A080355(n+1) is prime.

The prime p = 19 gives the prime 332887 = 1010001010001010111_2.

The positions of 1's from the end are 1, 2, 3, 5, 7, 11, 13, 17, 19.

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)).

Examples

			a(3) = 5 since 1 + 2^(2-1) + 2^(3-1) + 2^(5-1) = 10111_2 = 23 is prime.
Note that prime(3) = 5 and A080355(3+1) = 23 prime.
		

Crossrefs

Cf. A000040, A000043, A080355, A113878.

Programs

Mathematica

Prime@ Select[Range[10^3], PrimeQ[1 + Total@ Array[2^(Prime[#] - 1) &, #]] &] (* Michael De Vlieger, Oct 31 2018 *)

PARI

isok(p) = isprime(p) && isprime(1 + sum(k=1, primepi(p), 2^(prime(k)-1))); \\ Michel Marcus, Oct 27 2018

Extensions

a(5)-a(6) from Robert Israel, Oct 27 2018