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List of pairs (a,d) of coprime integers a>0, d>=0 such that a*(a+d)*(a+2*d) is a square, ordered by the squares.
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%I #17 Apr 06 2017 21:10:09

%S 1,0,1,24,18,7,1,840,49,120,50,119,49,240,128,161,98,527,289,336,800,

%T 41,162,1519,288,1081,529,840,1,28560,49,5280,961,720,289,2520,242,

%U 3479,49,9360,512,3713,529,3696,1568,1241,338,6887,2401,1320,2178,2047

%N List of pairs (a,d) of coprime integers a>0, d>=0 such that a*(a+d)*(a+2*d) is a square, ordered by the squares.

%C This is a 2-column table read by rows. For each row a,d the product a*(a+d)*(a+2*d) is a square. The rows are ordered by those products.

%C The main entry for this sequence is A284666, formed by the triples a, a+d, a+2*d. The square roots of the products a*(a+d)*(a+2*d) form A284876.

%C For a=1 the d values 0, 24, 840, 28560, ... are A078522.

%H Giovanni Resta, <a href="/A284874/b284874.txt">Table of n, a(n) for n = 1..832</a>

%F a(2*k+1) = A284666(3*k+1) and a(2*k+2) = A284666(3*k+2)-A284666(3*k+1) and a(2*k+1)*[a(2*k+1)+a(2*k+2)]*[a(2*k+1)+2*a(2*k+2)] = A284876(k+1)^2 for k>=0.

%e gcd(18,7)=1 and 18*(18+7)*(18+2*7) = 18*25*32 = 9*25*64 = (3*5*8)^2, so 18,7 is in the sequence.

%t nn = 50000; t = {};

%t p[a_, d_] := a (a + d) (a + 2 d); Do[

%t If[p[a, d] <= 2 nn^2 && GCD[a, d] == 1 && IntegerQ[Sqrt[p[a, d]]],

%t AppendTo[t, {a, d}]], {a, 1, nn}, {d, 0, nn}];

%t Sort[t, p[#1[[1]], #1[[2]]] < p[#2[[1]], #2[[2]]] &] // Flatten

%Y Cf. A078522, A284666, A284876.

%K nonn,tabf

%O 1,4

%A _Jonathan Sondow_, Apr 04 2017

%E a(37)-a(52) from _Giovanni Resta_, Apr 06 2017