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 A283354 #14 Apr 06 2017 21:06:38
%S A283354 5,11,281,15461,1194748781,636653,41,101,4007847353,
%T A283354 71544139714543458911,13553
%N A283354 Primes of the form 6k + 5 arising from an alternative proof that there are infinitely many primes of that form.
%C A283354 Start with a finite list of primes of the form 6k + 5, in this case, the 1-element set {5}. If the list has an even number of primes, duplicate one of them, preferably the smallest one. Then multiply the primes on the list (sometimes the larger primes will be multiplied by 25 rather than 5) and add 6.
%C A283354 Thus we get another number that is either a prime of the form 6k + 5 that it's not on our list, or a composite number that is the product of an odd number of primes of the form 6k + 5. Those primes are added to the list and the process can go through another iteration.
%C A283354 The classic proof that there are infinitely many primes of the form 6k + 5 uses a similar process, but the algorithm is indifferent to whether the finite list has an odd or even number of primes. We take the product of the primes, multiply by 6 and then subtract 1.
%e A283354 To start things off, let's say 5 is the only prime of the form 6k + 5.
%e A283354 But 5 + 6 = 11, which is also a prime of that form. So our list is now {5, 11}. Since that has an even number of terms, we temporarily amend the list to {5, 5, 11}.
%e A283354 Then 5^2 * 11 + 6 = 281, which is also a prime of that form. Our list is now {5, 11, 281}.
%e A283354 Then 5 * 11 * 281 = 15461, which is prime. Our list is now {5, 11, 281, 15461}. Since that has an even number of terms, we temporarily amend the list to {5, 5, 11, 281, 15461}.
%e A283354 Then 5^2 * 11 * 281 * 15461 + 6 = 1194748781, which is prime. Our list is now {5, 11, 281, 15461, 1194748781}.
%e A283354 Then 5 * 11 * 281 * 15461 * 1194748781 + 6 = 285484928506498661 = 636653 * 448415272537, of which the former is a prime of the form 6k + 5 and the latter is not. Our list is now {5, 11, 281, 15461, 1194748781, 636653}. Since that has an even number of terms, we temporarily amend the list to {5, 5, 11, 281, 15461, 1194748781, 636653}
%e A283354 Then 5^2 * 11 * 281 * 15461 * 1194748781 * 636653 + 6 = 908774180942239441008581 = 41 * 101 * 4007847353 * 54756991297, of which only the last factor is not of the form 6k + 5.
%Y A283354 Cf. A057206 (primes of the form 6k + 5 generated by the classic proof).
%K A283354 nonn,more
%O A283354 1,1
%A A283354 _Alonso del Arte_, Mar 05 2017