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%I #15 Jul 02 2021 10:38:25
%S 2,7,8,14,15,23,32,33,35,47,54,56,57,60,72,78,79,84,87,92,95,120,123,
%T 124,128,138,143,154,165,167,174,184,190,196,213,223,235,242,252,253,
%U 258,267,295,312,315,319,323,327,348,359,375,378,380,393,412,423,439
%N Numbers k such that sigma(k) + 1 is a square.
%H Harry J. Smith, <a href="/A063531/b063531.txt">Table of n, a(n) for n = 1..1000</a>
%F A000203(k) = -1 + m^2 for some m.
%e 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 Select[Range[500],IntegerQ[Sqrt[DivisorSigma[1,#]+1]]&] (* _Harvey P. Dale_, Jul 02 2021 *)
%o (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 Cf. A000010, A000203, A015722, A028871, A037074.
%K nonn
%O 1,1
%A _Labos Elemer_, Aug 02 2001
%E Minor edits from _Franklin T. Adams-Watters_, Aug 29 2009
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