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 A263770 #19 Dec 13 2023 08:38:38
%S A263770 7,5,7,17,13,29,19,41,73,31,97,191,43,89,97,109,61,311,137,73,149,241,
%T A263770 337,181,197,103,313,109,331,229,257,397,139,281,151,457,317,821,337,
%U A263770 349,181,547,193,389,199,401,1061,449,229,461,937,241,727,757,1033,1321,271,1361,557
%N A263770 Smallest prime q such that (prime(n)^2 + q*prime(n))/(prime(n) + 1) is an integer.
%C A263770 Least prime q such that q == 1 (mod prime(n) + 1).
%C A263770 Consists of the initial terms of sequences A002476, A002144, A002476, A007519, A068228, A140444, A061237, A141881, A107008, A132230, A133870, A141868, A124826, A142292, A142398, A141948, A088955, A142003, ...
%F A263770 5 is in this sequence because (prime(2)^2 + 5*prime(2))/(prime(2) + 1) = 6 and 5 is prime.
%t A263770 Table[q = 2; While[! IntegerQ[(Prime[n]^2 + q Prime@ n)/(Prime@ n + 1)], q = NextPrime@ q]; q, {n, 59}] (* _Michael De Vlieger_, Oct 26 2015 *)
%o A263770 (PARI) a(n) = {p = prime(n); q = 2; while ((p^2 + p*q) % (p + 1), q = nextprime(q+1)); q;} \\ _Michel Marcus_, Oct 26 2015
%Y A263770 Cf. A263729, A263730, A263769.
%K A263770 nonn
%O A263770 1,1
%A A263770 _Juri-Stepan Gerasimov_, Oct 25 2015