A111671 -id:A111671 - cp's OEIS frontend

A350696 a(n) is the least nonnegative integer k such that 2^n + k is a safe prime.

1, 3, 7, 15, 19, 39, 7, 51, 163, 15, 31, 231, 103, 75, 7, 195, 499, 99, 127, 627, 955, 555, 691, 87, 679, 99, 1411, 351, 799, 135, 91, 771, 79, 951, 667, 975, 1183, 1311, 667, 315, 955, 759, 2011, 9315, 4243, 1575, 907, 1527, 3943, 2091, 1927, 75, 1879

Offset: 2

Mark Andreas, Jan 12 2022

nonn

2^n+a(n) is the smallest (n+1)-bit safe prime.

			a(6)=19 because 2^6+19=83 is the smallest safe prime greater than 64 of the form p=2q+1 where p and q are both primes.

Mark Andreas, Table of n, a(n) for n = 2..5120

Cf. A111671, A057821, A243916, A181356, A335313.

safeQ[p_] := And @@ PrimeQ[{p, (p - 1)/2}]; a[n_] := Module[{k = 2^n + 1}, While[! safeQ[k], k++]; k -= 2^n]; Array[a, 50, 2] (* Amiram Eldar, Jan 12 2022 *)

a(n) = {my(k=0); until (isprime(2^n+k) && isprime((2^n+k-1)/2), k++); return (k); }

A367966 Smallest Sophie Germain prime >= 2^n.

Original entry on oeis.org

2, 2, 5, 11, 23, 41, 83, 131, 281, 593, 1031, 2063, 4211, 8243, 16421, 32771, 65633, 131321, 262193, 524351, 1048889, 2097629, 4194581, 8388953, 16777259, 33554771, 67108913, 134218433, 268435631, 536871311, 1073741891, 2147483693, 4294967681, 8589934631, 17179869659

Offset: 0

Andrei Lapets, Dec 06 2023

nonn

			For n = 0, a(0) = 2 because 2 is prime, 2*(2) + 1 = 5 is prime, 2 >= 2^0 where 2^0 = 1, and 1 is not prime.
For n = 1, a(1) = 2 because 2 is prime, 2*(2) + 1 = 5 is prime, 2 >= 2^1 where 2^1 = 2.
For n = 2, a(2) = 5 because 5 is prime, 2*(5) + 1 = 11 is prime, 5 >= 2^2 where 2^2 = 4, and 4 is not prime.

Robert Israel, Table of n, a(n) for n = 0..1000

Cf. A104080, A005384.

a:= proc(n) option remember; local p; for p from 2^n
      while not andmap(isprime, [p, 2*p+1]) do od; p
    end:
seq(a(n), n=0..44);  # Alois P. Heinz, Dec 13 2023

a={}; nmax=35; For[n=0, n<=nmax, n++, k=2^n; While[!PrimeQ[k] || !PrimeQ[2k+1], k++]; AppendTo[a,k]]; a (* Stefano Spezia, Dec 10 2023 *)

a(n) = forprime(p=2^n, , if (isprime(2*p+1), return(p))); \\ Michel Marcus, Dec 12 2023

Apparently a(n) = (A111671(n) - 1)/2 for n>=2. - Hugo Pfoertner, Dec 13 2023

cp's OEIS Frontend

A350696 a(n) is the least nonnegative integer k such that 2^n + k is a safe prime.

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