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%I #7 Feb 16 2024 14:57:49
%S 0,20,0,70,56,162,0,160,308,110,324,520,0,286,560,810,182,540,880,
%T 1190,0,448,884,1296,1672,272,810,1330,1820,0,646,2268,1280,1890,2464,
%U 380,1134,2990,1870,2576,3240,0,880,1748,2592,3850,3400,506,1512
%N Values d for the discriminant d^2 = 4p^3 - 27q*2 of the depressed cubic equation x^3 - p*x + q = 0 that give integer roots using integer coefficients p > 0 and q > 0 for increasing p sorted by p then q.
%C To obtain integer roots from the depressed cubic x^3 - p*x + q = 0, its discriminant 4p^3 - 27q*2 has to be a perfect square but this is not a sufficient condition. At least one root has to be integral as well.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Discriminant#Degree_3">Discriminant - Degree 3</a>.
%e a(3) = 70 and occurs when (p, q) = (13, 12). The depressed cubic is given as x*3 - 13x + 12 and has roots (-4, 1, 3}. It is the 3rd occurrence of a solution set of 3 integers.
%t lst = {}; Do[If[IntegerQ[k=(4p^3-27q^2)^(1/2)], (sol=Solve[x^3-p*x+q==0, {x}]; {x1, x2, x3}=x /. sol; If[IntegerQ[x1], AppendTo[lst, k]])], {p, 1, 300}, {q, 1, Sqrt[4 p^3/27]}]; lst
%Y Cf. A082375.
%K nonn
%O 1,2
%A _Frank M Jackson_, Feb 14 2024