A275767 Numbers k for which 2*4^k - 27 is prime.

2, 3, 9, 11, 291, 1263, 2661, 3165, 8973, 8999, 27479

Offset

1,1

Comments

The prime numbers that these exponents generate are given in A275749.

Since 2*4^(2k) - 27 = 2*16^k - 27 == (2*1^k - 27) mod 5 = -25 mod 5 == 0 mod 5, no even number greater than 2 will be in this sequence.

a(8) > 5000. - Vincenzo Librandi, Aug 08 2016

Links

Table of n, a(n) for n=1..11.

D. Alpern, Factorization using the Elliptic Curve Method.

Example

a(1) = 2, since 2*4^2 - 27 = 32 - 27 = 5, which is prime.
a(2) = 3, since 2*4^3 - 27 = 128 - 27 = 101, which is prime.
a(3) = 9, since 2*4^9 - 27 = 524288 - 27 = 524261, which is prime.
a(4) = 11, since 2*4^11 - 27 = 8388608 - 27 = 8388581, which is prime.

Mathematica

Select[Range[2, 1000], PrimeQ[2 4^# - 27] &] (* Vincenzo Librandi, Aug 08 2016 *)

Prog

(Magma) [n: n in [2..1000] |IsPrime(2*4^n-27)]; // Vincenzo Librandi, Aug 08 2016
(Python)
from sympy import isprime
def afind(limit, startk=2):
    alst, pow4 = [], 4**startk
    for k in range(startk, limit+1):
        if isprime(2*pow4 - 27): print(k, end=", ")
        pow4 *= 4
afind(1300) # Michael S. Branicky, Sep 22 2021

Crossrefs

Cf. A274519, A275749.

Sequence in context: A214259 A287680 A242680 * A088086 A088084 A182203

Adjacent sequences: A275764 A275765 A275766 * A275768 A275769 A275770

Keyword

nonn,more

Author

Timothy L. Tiffin, Aug 07 2016

Extensions

a(6)-a(8) from Vincenzo Librandi, Aug 08 2016

a(9)-a(10) from Michael S. Branicky, Sep 22 2021

a(11) from Michael S. Branicky, Apr 05 2023

Status

approved