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 A246119 #69 Jun 01 2023 03:27:12
%S A246119 2,2,2,5,4,2,5,196,14,129,424,484,22,5164,7726,13325,96873,192098,
%T A246119 712012,123447
%N A246119 a(n) is the least k such that k^(2^n)*(k^(2^n) - 1) + 1 is prime.
%C A246119 Numbers of the form k^m*(k^m-1)+1 with m > 0, k > 1 may be primes only if m is 3-smooth, because these numbers are Phi(6,k^m) and cyclotomic factorizations apply to any prime divisors >3. This series is a subset of A205506 with only m=2^n.
%C A246119 Trivially, a(n) <= a(n+1)^2. This upper bound, indeed, holds for a(4) = a(5)^2, a(7) = a(8)^2 and a(11) = a(12)^2.
%C A246119 The numbers of this form are Generalized Unique primes (see Links section).
%C A246119 a(16)=96873 corresponds to a prime with 653552 decimal digits.
%C A246119 The search for a(17) which corresponds to a 1385044-decimal digit prime was performed on a small Amazon EC2 cloud farm (40 GRID K520 GPUs), at a cost of approximately $1000 over three weeks.
%C A246119 a(18) <= 712012 corresponds to a prime with 3068389 decimal digits. - _Serge Batalov_, Jan 15 2018
%C A246119 a(19) <= 123447 corresponds to a prime with 5338805 decimal digits. - _Serge Batalov_, Jan 15 2018
%C A246119 a(20) <= 465859 corresponds to a prime with 11887192 decimal digits (not all lower candidates have been checked). This is the largest known non-Mersenne prime at the time of its discovery. - _Serge Batalov_, May 31 2023
%H A246119 C. Caldwell, <a href="https://t5k.org/top20/page.php?id=44">Generalized unique primes</a>
%H A246119 C. Caldwell, The Prime Pages, <a href="https://t5k.org/primes/page.php?id=136107">Phi_3(-465859^1048576)</a>
%F A246119 a(n) = A085398(3*2^(n+1)). - _Jinyuan Wang_, Jan 01 2023
%t A246119 Table[SelectFirst[Range@ 200, PrimeQ[#^(2^n) (#^(2^n) - 1) + 1] &], {n, 0, 9}] (* _Michael De Vlieger_, Jan 15 2018 *)
%o A246119 (PARI)
%o A246119 a(n)=k=1;while(!ispseudoprime(k^(2^n)*(k^(2^n)-1)+1),k++);k
%o A246119 n=0;while(n<100,print1(a(n),", ");n++) \\ _Derek Orr_, Aug 14 2014
%Y A246119 Cf. A056993, A085398, A101406, A153436, A153438, A205506, A246120, A246121, A298206.
%K A246119 nonn,more,hard
%O A246119 0,1
%A A246119 _Serge Batalov_, Aug 14 2014
%E A246119 a(16) from _Serge Batalov_, Dec 30 2014
%E A246119 a(17) from _Serge Batalov_, Feb 10 2015
%E A246119 a(18-19) from _Serge Batalov_, May 31 2023