login

Reminder: The OEIS is hiring a new managing editor, and the application deadline is January 26.

Numbers k for which 2*4^k - 27 is prime.
3

%I #29 Oct 25 2024 12:36:54

%S 2,3,9,11,291,1263,2661,3165,8973,8999,27479,42689

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

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

%C 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.

%C a(8) > 5000. - _Vincenzo Librandi_, Aug 08 2016

%H D. Alpern, <a href="https://www.alpertron.com.ar/ECM.HTM">Factorization using the Elliptic Curve Method</a>.

%e a(1) = 2, since 2*4^2 - 27 = 32 - 27 = 5, which is prime.

%e a(2) = 3, since 2*4^3 - 27 = 128 - 27 = 101, which is prime.

%e a(3) = 9, since 2*4^9 - 27 = 524288 - 27 = 524261, which is prime.

%e a(4) = 11, since 2*4^11 - 27 = 8388608 - 27 = 8388581, which is prime.

%t Select[Range[2, 1000], PrimeQ[2 4^# - 27] &] (* _Vincenzo Librandi_, Aug 08 2016 *)

%o (Magma) [n: n in [2..1000] |IsPrime(2*4^n-27)]; // _Vincenzo Librandi_, Aug 08 2016

%o (Python)

%o from sympy import isprime

%o def afind(limit, startk=2):

%o alst, pow4 = [], 4**startk

%o for k in range(startk, limit+1):

%o if isprime(2*pow4 - 27): print(k, end=", ")

%o pow4 *= 4

%o afind(1300) # _Michael S. Branicky_, Sep 22 2021

%Y Cf. A274519, A275749.

%K nonn,more

%O 1,1

%A _Timothy L. Tiffin_, Aug 07 2016

%E a(6)-a(8) from _Vincenzo Librandi_, Aug 08 2016

%E a(9)-a(10) from _Michael S. Branicky_, Sep 22 2021

%E a(11) from _Michael S. Branicky_, Apr 05 2023

%E a(12) from _Michael S. Branicky_, Oct 25 2024