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 A261382 #11 Aug 17 2015 14:29:38
%S A261382 2,2510,15,30,5,510,730,440,195,6230,2040,2760,20,1010,12570,31340,
%T A261382 1625,1650,725,2480,2160,520,1055,60,5,20,1260,25800,6185,6240,10,
%U A261382 1180,12600,7500,5330,390,325,2880,11655,32670,5850,43110,3230,1470,7680,4950,255,202650,10530,450,2445,11670,8745,103350,80,6890,135,18930,80,245040
%N A261382 Least positive integer k such that (k-1)^2+(k*n)^2, k^2+(k*n-1)^2, (k+1)^2+(k*n)^2 and k^2+(k*n+1)^2 are all prime.
%C A261382 Conjecture: a(n) exists for any n > 0. In general, any positive rational number r can be written as m/n, where m and n are positive integers with (m-1)^2+n^2, m^2+(n-1)^2, (m+1)^2+n^2 and m^2+(n+1)^2 all prime.
%C A261382 It is easy to prove that if m and n are positive integers with (m-1)^2+n^2, m^2+(n-1)^2, (m+1)^2+n^2 and m^2+(n+1)^2 all prime, then either m = n = 2 or m == n == 0 (mod 5).
%D A261382 Zhi-Wei Sun, Problems on combinatorial properties of primes, in: M. Kaneko, S. Kanemitsu and J. Liu (eds.), Number Theory: Plowing and Starring through High Wave Forms, Proc. 7th China-Japan Seminar (Fukuoka, Oct. 28 - Nov. 1, 2013), Ser. Number Theory Appl., Vol. 11, World Sci., Singapore, 2015, pp. 169-187.
%H A261382 Zhi-Wei Sun, <a href="/A261382/b261382.txt">Table of n, a(n) for n = 1..1000</a>
%H A261382 Zhi-Wei Sun, <a href="/A261382/a261382.txt">Checking the conjecture for r = a/b with a,b = 1..60</a>
%H A261382 Zhi-Wei Sun, <a href="http://arxiv.org/abs/1402.6641">Problems on combinatorial properties of primes</a>, arXiv:1402.6641 [math.NT], 2014.
%e A261382 a(2) = 2510 since (2510-1)^2+(2510*2)^2 = 31495481, 2510^2+(2510*2-1)^2 = 31490461, (2510+1)^2+(2510*2)^2 = 31505521 and 2510^2+(2510*2+1)^2 = 31510541 are all prime.
%t A261382 PQ[p_]:=PrimeQ[p]
%t A261382 q[m_,n_]:=PQ[(m-1)^2+n^2]&&PQ[m^2+(n-1)^2]&&PQ[(m+1)^2+n^2]&&PQ[m^2+(n+1)^2]
%t A261382 Do[k=0;Label[bb];k=k+1;If[q[k,k*n],Goto[aa],Goto[bb]];Label[aa];Print[n," ", k];Continue,{n,1,60}]
%o A261382 (PARI) is_ok(k,n)=isprime((k-1)^2+(k*n)^2)&&isprime(k^2+(k*n-1)^2)&&isprime((k+1)^2+(k*n)^2)&&isprime(k^2+(k*n+1)^2)
%o A261382 first(m)=my(v=vector(m),k=1);for(i=1,m,while(!is_ok(k,i),k++);v[i]=k;k++;);v; \\ _Anders Hellström_, Aug 17 2015
%Y A261382 Cf. A000040, A000290, A236193, A259492, A261339.
%K A261382 nonn
%O A261382 1,1
%A A261382 _Zhi-Wei Sun_, Aug 17 2015