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 A063531 #23 Jul 10 2024 02:57:34
%S A063531 2,7,8,14,15,23,32,33,35,47,54,56,57,60,72,78,79,84,87,92,95,120,123,
%T A063531 124,128,138,143,154,165,167,174,184,190,196,213,223,235,242,252,253,
%U A063531 258,267,295,312,315,319,323,327,348,359,375,378,380,393,412,423,439
%N A063531 Numbers k such that sigma(k) + 1 is a square.
%C A063531 Numbers k such that A000203(k) = -1 + m^2 for some m.
%H A063531 Amiram Eldar, <a href="/A063531/b063531.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Harry J. Smith)
%e A063531 If k = p(p+2) is a product of twin primes (from A037074), then sigma(k) + 1 = 1 + (p+1)(p+3) = (p+2)^2, square of the larger twin. Other solutions can be either special primes = m^2 - 2 or composites like 120: sigma(120) = 120 + 60 + ... + 1 = 360 = 19^2 - 1. Square number solution is, e.g., 196: sigma(196) = 399 = 20^2 - 1.
%t A063531 Select[Range[500],IntegerQ[Sqrt[DivisorSigma[1,#]+1]]&] (* _Harvey P. Dale_, Jul 02 2021 *)
%o A063531 (PARI) { n=0; for (a=1, 10^9, if (issquare(sigma(a) + 1), write("b063531.txt", n++, " ", a); if (n==1000, break)) ) } \\ _Harry J. Smith_, Aug 25 2009
%Y A063531 Cf. A000203, A028871, A037074, A063530, A088580.
%K A063531 nonn
%O A063531 1,1
%A A063531 _Labos Elemer_, Aug 02 2001
%E A063531 Minor edits from _Franklin T. Adams-Watters_, Aug 29 2009