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 A309499 #14 Aug 05 2019 17:54:44
%S A309499 2,11,17,19,283,919,1093,1787,9521,181243,257611,274243,857419,
%T A309499 1644871,3111607,6027277,10452083,14490703,36102991,47352131,
%U A309499 121431767,171236887,339934099,584698243,1177972427,3008777311,3091999399
%N A309499 Primes p having a record value of least k such that 4*k^2*p^2 + 1 is prime.
%C A309499 The corresponding record values of k are 1, 3, 5, 15, 20, 22, 24, 45, 95, 104, 115, 116, 135, 143, 155, 165, 179, 186, 190, 245, 250, 260, 277, 284, 310, 313, 335, ...
%C A309499 Gagola calculated the values of k for all the primes below 5000, and noticed that the largest value of k was only 45.
%H A309499 Gloria Gagola, <a href="https://www.jstor.org/stable/2689383">Progress on primes</a>, News and Letters, Mathematics Magazine, Vol. 54, No. 1 (1981), p. 43.
%e A309499 For the primes p = 2, 3, 5, and 7, 4*p^2 + 1 = 17, 37, 101, and 197 are all primes with k = 1. 11 is the first prime with a value of k = 3, since 4*1*11^2 + 1 = 45 and 4*2^2*11^2 + 1 = 1937 are both composites, and 4*3^2*11^2 + 1 = 4357 is prime.
%t A309499 a[p_] := Module[{k = 1}, While[!PrimeQ[4 * k^2 * p^2 + 1], k++]; k]; s={}; am = 0; p = 1; Do[p = NextPrime[p]; a1 = a[p]; If[a1 > am, am=a1; AppendTo[s, p]], {n, 1, 20000}]; s
%Y A309499 Cf. A309498.
%K A309499 nonn,more
%O A309499 1,1
%A A309499 _Amiram Eldar_, Aug 05 2019