This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A387060 #69 Sep 02 2025 23:38:13
%S A387060 0,3,4,5,12,24,36,77,195,296,297,533,545,644,884,932,1409,2061,2453,
%T A387060 2985,3381,4980,5393,11733,13631,14516,21004,27663,32645,39453,67055,
%U A387060 90543
%N A387060 Numbers k such that 16 * 3^k + 1 is prime.
%C A387060 a(33) > 10^5.
%C A387060 Conjecture: The only intersection with A385115 is at k = 3 where 2^4 * 3^3 = 432 = A027856(8).
%C A387060 Idea: For odd k > 3, covering systems ensure mutual exclusion:
%C A387060 If k = 1, 9, 13, 19, 25, 31, 37, 39, 43, 49, 55 (mod 60), then 7 or 31 divides (16*3^k+1).
%C A387060 If k = 5, 7, 11, 17, 23, 27, 29, 35, 41, 47, 53, 57, 59 (mod 60), then 11 or 13 divides (16*3^k-1).
%C A387060 If k = 15, 21, 33, 45, 51 (mod 60), various primes including {11,31,43,109,277,433,...} ensure at least one of 16*3^k +- 1 is composite.
%C A387060 If k = 3 (mod 60) and k > 3, the probability of intersection becomes vanishingly small.
%C A387060 Only k = 3 escapes all divisibility conditions. Verified to k = 10^5.
%H A387060 Ken Clements, <a href="/A387060/a387060_2.py.txt">Python program to calculate covering system.</a>
%t A387060 Select[Range[0, 4000], PrimeQ[16*3^# + 1] &] (* _Amiram Eldar_, Aug 16 2025 *)
%o A387060 (Python)
%o A387060 from gmpy2 import is_prime
%o A387060 print([k for k in range(4_000) if is_prime(16 * 3**k + 1)])
%Y A387060 Cf. A027856, A005537, A005538, A385115.
%K A387060 nonn,more,new
%O A387060 1,2
%A A387060 _Ken Clements_, Aug 15 2025