A123250 Primes of the form 2^k + 5.

7, 13, 37, 2053, 140737488355333, 9007199254740997, 2787593149816327892691964784081045188247557, 11150372599265311570767859136324180752990213, 3138550867693340381917894711603833208051177722232017256453

Offset: 1

Author

Cino Hilliard, Oct 08 2006

Keywords

• nonn

Comments

A059242 is the main entry for this sequence.

If 2^n + 5 is prime then n is odd. Proof: Lemma 1: a^n+b^n = (a+b)(a^n-1 - a^(n-2)b + ... + b^(n-1)) 2^n + 5 = 2*(2^(n-1)+1) + 3. Then if n is even, n-1 is odd and by Lemma 1, 2+1 divides 2*(2^(n-1)+1) and thus divides 2^n+5 so it cannot be prime.

Crossrefs

Cf. A000040, A059242.

Programs

Mathematica

Select[Table[2^k+5,{k,200}],PrimeQ] (* James C. McMahon, Nov 19 2024 *)

PARI

g(n,p) = for(k=1,n,y=p+2^k;if(isprime(y),print1(y",")))

Formula

a(n) = 2^A059242(n) + 5. - Elmo R. Oliveira, Nov 08 2023

Extensions

a(9) from James C. McMahon, Nov 19 2024