cp's OEIS Frontend

A274519 Numbers k for which 4^k - 27 is prime.

%I A274519 #72 Oct 25 2024 12:36:22
%S A274519 3,4,5,10,11,13,25,28,29,65,70,115,305,515,2029,2393,2605,3530,4036,
%T A274519 4750,10288,11048,11596,29359,32123,47371
%N A274519 Numbers k for which 4^k - 27 is prime.
%C A274519 The prime numbers that these exponents generate are given in A275750.
%C A274519 Since 4^(6k) - 27 = 4096^k - 27 == (1^k - 27) mod 13 = -26 mod 13 == 0 mod 13, no multiple of 6 will be in this sequence. Also, since 4^(5k+2) - 27 = 16*1024^k - 27 == (16*1^k - 27) mod 11 = -11 mod 11 == 0 mod 11, no number congruent to 2 mod 5 will be in this sequence.
%C A274519 a(21) > 5000. - _Vincenzo Librandi_, Aug 08 2016
%H A274519 D. Alpern, <a href="https://www.alpertron.com.ar/ECM.HTM">Factorization using the Elliptic Curve Method</a>.
%e A274519 a(1) = 3, since 4^3 - 27 = 64 - 27 = 37, which is prime.
%e A274519 a(2) = 4, since 4^4 - 27 = 256 - 27 = 229, which is prime.
%e A274519 a(3) = 5, since 4^5 - 27 = 1024 - 27 = 997, which is prime.
%e A274519 a(4) = 10, since 4^10 - 27 = 1048576 - 27 = 1048549, which is prime.
%e A274519 a(5) = 11, since 4^11 - 27 = 4194304 - 27 = 4194277, which is prime.
%e A274519 a(6) = 13, since 4^13 - 27 = 67108864 - 27 = 67108837, which is prime.
%t A274519 Select[Range[3, 5000], PrimeQ[4^# - 27] &] (* _Vincenzo Librandi_, Aug 08 2016 *)
%o A274519 (Magma) [n: n in [3..2000] |IsPrime(4^n-27)]; // _Vincenzo Librandi_, Aug 08 2016
%o A274519 (Python)
%o A274519 from sympy import isprime
%o A274519 def afind(limit, startk=3):
%o A274519     alst, pow4 = [], 4**startk
%o A274519     for k in range(startk, limit+1):
%o A274519         if isprime(pow4 - 27): print(k, end=", ")
%o A274519         pow4 *= 4
%o A274519 afind(600) # _Michael S. Branicky_, Sep 22 2021
%Y A274519 Cf. A275750, A275767.
%K A274519 nonn,more
%O A274519 1,1
%A A274519 _Timothy L. Tiffin_, Aug 07 2016
%E A274519 a(13)-a(20) from _Vincenzo Librandi_, Aug 08 2016
%E A274519 a(21)-a(23) from _Michael S. Branicky_, Sep 22 2021
%E A274519 a(24)-a(26) from _Michael S. Branicky_, Oct 25 2024