OFFSET
1,1
COMMENTS
The prime numbers that these exponents generate are given in A275750.
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.
a(21) > 5000. - Vincenzo Librandi, Aug 08 2016
LINKS
EXAMPLE
a(1) = 3, since 4^3 - 27 = 64 - 27 = 37, which is prime.
a(2) = 4, since 4^4 - 27 = 256 - 27 = 229, which is prime.
a(3) = 5, since 4^5 - 27 = 1024 - 27 = 997, which is prime.
a(4) = 10, since 4^10 - 27 = 1048576 - 27 = 1048549, which is prime.
a(5) = 11, since 4^11 - 27 = 4194304 - 27 = 4194277, which is prime.
a(6) = 13, since 4^13 - 27 = 67108864 - 27 = 67108837, which is prime.
MATHEMATICA
Select[Range[3, 5000], PrimeQ[4^# - 27] &] (* Vincenzo Librandi, Aug 08 2016 *)
PROG
(Magma) [n: n in [3..2000] |IsPrime(4^n-27)]; // Vincenzo Librandi, Aug 08 2016
(Python)
from sympy import isprime
def afind(limit, startk=3):
alst, pow4 = [], 4**startk
for k in range(startk, limit+1):
if isprime(pow4 - 27): print(k, end=", ")
pow4 *= 4
afind(600) # Michael S. Branicky, Sep 22 2021
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Timothy L. Tiffin, Aug 07 2016
EXTENSIONS
a(13)-a(20) from Vincenzo Librandi, Aug 08 2016
a(21)-a(23) from Michael S. Branicky, Sep 22 2021
a(24)-a(26) from Michael S. Branicky, Oct 25 2024
STATUS
approved