A057673 -id:A057673 - cp's OEIS frontend

A056206 Smallest prime p such that p + 2^n is also a prime.

2, 3, 3, 3, 3, 5, 3, 3, 7, 11, 7, 5, 3, 17, 37, 3, 3, 29, 3, 53, 7, 17, 67, 11, 43, 41, 97, 29, 3, 11, 3, 11, 61, 17, 79, 53, 31, 29, 7, 23, 97, 71, 277, 29, 7, 59, 127, 5, 61, 191, 193, 101, 37, 5, 163, 3, 97, 131, 577, 131, 151, 197, 193, 29, 13, 131, 709, 3, 61

Offset: 0

Labos Elemer, Oct 06 2000

nonn

			n=9, 512 + {2,3,5,7,11,...} = {514,515,519,523,...} = {2*257, 5*103, 11*47, 3*173, 523=prime, ...}. The smallest suitable prime is 11 and it gives 523 = 512 + 11. So a(9)=11.

Alois P. Heinz, Table of n, a(n) for n = 0..2500 (first 1001 terms from T. D. Noe)

Cf. A056208, A057673.

a:= proc(n) option remember; local o, p; o, p:=2^n, 2;
      while not isprime(o+p) do p:= nextprime(p) od; p
    end:
seq(a(n), n=0..75);  # Alois P. Heinz, Mar 20 2023

Table[i=1; While[!PrimeQ[2^n+(p=Prime[i])],i++]; p,{n,0,72}] (* Jayanta Basu, May 23 2013 *)

a(n) = Min{p|p+2^n=q, both p and q are primes}.

a(0) from Jayanta Basu, May 23 2013

A381041 Smallest prime p such that 3^n + p + 1 is prime.

Original entry on oeis.org

3, 3, 3, 3, 7, 7, 3, 19, 7, 3, 3, 19, 79, 7, 7, 43, 67, 139, 127, 103, 7, 97, 3, 31, 31, 13, 379, 61, 109, 433, 3, 79, 127, 79, 67, 139, 127, 229, 7, 109, 271, 313, 3, 151, 7, 103, 67, 283, 421, 67, 43, 373, 97, 97, 97, 19, 61, 3, 157, 331, 127, 37, 139, 439, 421

Offset: 0

James S. DeArmon, Apr 20 2025

nonn

			a(0) = 3, since 3 + (3^0+1) = 5 is prime and 2 + (3^0+1) = 4 is not.
a(1) = 3, since 3 + (3^1+1) = 7 is prime and 2 + (3^1+1) = 6 is not.

Robert Israel, Table of n, a(n) for n = 0..2000

Cf. A007051, A020483, A056206, A056208, A057673.

f:= proc(n) local t,p;
 p:= 1: t:= 3^n+1;
 do
   p:= nextprime(p);
   if isprime(p+t) then return p fi
 od;
end proc:
map(f, [$0..100]); # Robert Israel, Jun 19 2025

a[n_]:=Module[{p=2},While[!PrimeQ[p+3^n+1], p=NextPrime[p]]; p]; Array[a,65,0] (* Stefano Spezia, Apr 25 2025 *)

a(n) = my(p=2, x=3^n+1); while (!isprime(p+x), p=nextprime(p+1)); p; \\ Michel Marcus, Apr 24 2025

from sympy import isprime, nextprime
def a(n):
    p, b = 2, 3**n+1
    while not isprime(p+b):
        p = nextprime(p)
    return p
print([a(n) for n in range(65)]) # Michael S. Branicky, Apr 23 2025

from sympy import nextprime, isprime
def A381041(n):
    p = 3**n+1
    q = nextprime(p)
    while not isprime(q-p):
        q = nextprime(q)
    return q-p # Chai Wah Wu, May 01 2025

a(n) = A020483(A007051(n)). - Robert Israel, Jun 19 2025

More terms from Michael S. Branicky, Apr 23 2025

A295111 Primes p such that 2^p - p is also a prime.

Original entry on oeis.org

2, 3, 13, 19, 481801

Offset: 1

Iain Fox, Nov 14 2017

a(6) > 1061095.
Intersection of A000040 and A048744.
Since numbers other than 3 that are congruent to 3 mod 6 are composite, for n > 2, a(n) is congruent to 1 mod 6 (see comments by Iain Fox in A048744).

			p=13, 2^13 - 13 = 8179 is prime.

Cf. A000325, A081296, A048744, A057663, A057673.

lista(nn) = forprime(p=2, nn, if(ispseudoprime(2^p - p), print1(p, ", ")))

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A056206 Smallest prime p such that p + 2^n is also a prime.

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A381041 Smallest prime p such that 3^n + p + 1 is prime.

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A295111 Primes p such that 2^p - p is also a prime.

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PARI