A181628 Numbers k such that (2^k + 3^k)/13 is prime.

6, 10, 14, 22, 34, 38, 82, 106, 218, 334, 4414, 7246, 10118, 10942, 15898, 42422, 65986

Offset: 1

Author

Michel Lagneau, Nov 18 2010

Keywords

• nonn
• more

Comments

All terms are of the form 2p, p prime.

The prime (2^4414 + 3^4414)/13 = 79300327387 ...611266 985181 has 2105 decimal digits.

a(18) > 10^5. - Michael S. Branicky, Aug 17 2024

Examples

			10 is in the sequence because (2^10+ 3^10)/13 = 60073/13 = 4621 is prime.
		

Crossrefs

Cf. A057469.

Programs

Maple

with(numtheory):for n from 1 to 4500 do: x:= (2^n + 3^n)/13:if floor(x)=x and
  type(x,prime)=true then printf(`%d, `, n):else fi:od:
# alternative
Res:= NULL:
p:= 2:
while p < 6000 do
p:= nextprime(p);
if isprime((2^(2*p)+3^(2*p))/13) then Res:= Res, 2*p fi;
od:
Res; # Robert Israel, Apr 26 2017

PARI

is(n)=n%2==0 && isprime(n/2) && ispseudoprime((2^n+3^n)/13) \\ Charles R Greathouse IV, Jun 06 2017

Python

from sympy import isprime
def afind(limit, startk=1):
    k = startk
    pow2 = 2**k
    pow3 = 3**k
    for k in range(startk, limit+1):
        q, r = divmod(pow2+pow3, 13)
        if r == 0 and isprime(q):
            print(k, end=", ")
        pow2 *= 2
        pow3 *= 3
afind(1000) # Michael S. Branicky, Dec 28 2021

Extensions

a(12) from D. S. McNeil, Nov 18 2010

a(13) and a(14) from Robert Israel, Apr 26 2017

a(15) from Michael S. Branicky, Dec 28 2021

a(16) from Michael S. Branicky, Apr 26 2023

a(17) from Michael S. Branicky, Aug 17 2024