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

%I A275767 #29 Oct 25 2024 12:36:54
%S A275767 2,3,9,11,291,1263,2661,3165,8973,8999,27479,42689
%N A275767 Numbers k for which 2*4^k - 27 is prime.
%C A275767 The prime numbers that these exponents generate are given in A275749.
%C A275767 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 A275767 a(8) > 5000. - _Vincenzo Librandi_, Aug 08 2016
%H A275767 D. Alpern, <a href="https://www.alpertron.com.ar/ECM.HTM">Factorization using the Elliptic Curve Method</a>.
%e A275767 a(1) = 2, since 2*4^2 - 27 = 32 - 27 = 5, which is prime.
%e A275767 a(2) = 3, since 2*4^3 - 27 = 128 - 27 = 101, which is prime.
%e A275767 a(3) = 9, since 2*4^9 - 27 = 524288 - 27 = 524261, which is prime.
%e A275767 a(4) = 11, since 2*4^11 - 27 = 8388608 - 27 = 8388581, which is prime.
%t A275767 Select[Range[2, 1000], PrimeQ[2 4^# - 27] &] (* _Vincenzo Librandi_, Aug 08 2016 *)
%o A275767 (Magma) [n: n in [2..1000] |IsPrime(2*4^n-27)]; // _Vincenzo Librandi_, Aug 08 2016
%o A275767 (Python)
%o A275767 from sympy import isprime
%o A275767 def afind(limit, startk=2):
%o A275767     alst, pow4 = [], 4**startk
%o A275767     for k in range(startk, limit+1):
%o A275767         if isprime(2*pow4 - 27): print(k, end=", ")
%o A275767         pow4 *= 4
%o A275767 afind(1300) # _Michael S. Branicky_, Sep 22 2021
%Y A275767 Cf. A274519, A275749.
%K A275767 nonn,more
%O A275767 1,1
%A A275767 _Timothy L. Tiffin_, Aug 07 2016
%E A275767 a(6)-a(8) from _Vincenzo Librandi_, Aug 08 2016
%E A275767 a(9)-a(10) from _Michael S. Branicky_, Sep 22 2021
%E A275767 a(11) from _Michael S. Branicky_, Apr 05 2023
%E A275767 a(12) from _Michael S. Branicky_, Oct 25 2024