A367966 Smallest Sophie Germain prime >= 2^n.
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
Keywords
Examples
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.
Links
- Robert Israel, Table of n, a(n) for n = 0..1000
Programs
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Maple
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 -
Mathematica
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 *) -
PARI
a(n) = forprime(p=2^n, , if (isprime(2*p+1), return(p))); \\ Michel Marcus, Dec 12 2023
Formula
Apparently a(n) = (A111671(n) - 1)/2 for n>=2. - Hugo Pfoertner, Dec 13 2023