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 A353214 #44 May 09 2022 15:12:19
%S A353214 0,-1,-2,2,-4,-2,-8,2,-5,-2,4,-2,5,2,-11,-20,-22,6,-23,-21,2,-3,-16,
%T A353214 -25,-31,40,19,-29,-2,-2,2,-49,19,68,-56,-23,-59,45,29,-2,62,63,27,54,
%U A353214 -2,-22,-46,28,-85,-2,-29,17,-113,-4,-128,-65,-46,20,-51,-98,-64
%N A353214 a(n) = 2^A007013(4) mod prime(n); the last term of this sequences is when a(n) = 1.
%C A353214 This sequence uses the centered version of mod. The residue system modulo prime(n) is {-1*floor(prime(n)/2)..floor(prime(n)/2)}. This is so that this sequence will encode information about the numbers around 2^A007013(4). If a(n) = k and prime(n) < 2^A007013(4) - k, then 2^A007013(4) - k is not prime (prime(n) is a factor of 2^A007013(4) - k). For example, a(22) = -3, so prime(22) = 79 is a factor of 2^A007013(4) + 3.
%C A353214 The length of this sequence is the lowest value of n such that A014664(n) = A007013(4). This is because for any power of 2, 2^p, if p == 0 (mod A014664(n)), then 2^p == 1 (mod prime(n)) (prime(n) is a factor of A000225(p)). Since A007013(4) is prime, we can apply this to get: If A014664(n) = A007013(4) and prime(n) < A007013(5), then A007013(5) is not prime (prime(n) is a nontrivial factor).
%C A353214 For any n such that prime(n) < 5*(10^51 + 5*10^9), a(n) != 1.
%H A353214 Chris K. Caldwell, <a href="https://primes.utm.edu/mersenne/#unknown">Mersenne Primes: History, Theorems and Lists (5. Conjectures and Unsolved Problems)</a>.
%H A353214 I. S. Eum, <a href="https://doi.org/10.1007/s13226-018-0281-8">A congruence relation of the Catalan-Mersenne numbers</a>, Indian J Pure Appl Math, 49 (2018), 521-526.
%H A353214 Robert Delion, <a href="https://www.researchgate.net/publication/291336543_The_n2_1_Fermat_and_Mersenne_prime_numbers_conjectures_are_resolved">The n2 + 1 Fermat and Mersenne prime numbers conjectures are resolved</a>, Theoretical Mathematics & Applications, vol.6, no.1, 2016, 15-37.
%F A353214 a(n) = 2^(2^127 - 1) mod prime(n).
%o A353214 (PARI) A353214(n)=my(CM4=shift(1,127)-1);centerlift(Mod(2,prime(n))^CM4)
%Y A353214 Cf. A000225, A007013, A014664. Powers of 2 mod primes: A201908, A201912, A353171.
%K A353214 sign,fini
%O A353214 1,3
%A A353214 _Davis Smith_, Apr 30 2022