A284874 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.

1, 0, 1, 24, 18, 7, 1, 840, 49, 120, 50, 119, 49, 240, 128, 161, 98, 527, 289, 336, 800, 41, 162, 1519, 288, 1081, 529, 840, 1, 28560, 49, 5280, 961, 720, 289, 2520, 242, 3479, 49, 9360, 512, 3713, 529, 3696, 1568, 1241, 338, 6887, 2401, 1320, 2178, 2047

Offset: 1

Author

Jonathan Sondow, Apr 04 2017

Keywords

• nonn
• tabf

Comments

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.

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.

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

Examples

			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.
		

Links

• Giovanni Resta, Table of n, a(n) for n = 1..832

Crossrefs

Cf. A078522, A284666, A284876.

Programs

Mathematica

nn = 50000; t = {};
p[a_, d_] := a (a + d) (a + 2 d); Do[
If[p[a, d] <= 2 nn^2 && GCD[a, d] == 1 && IntegerQ[Sqrt[p[a, d]]],
  AppendTo[t, {a, d}]], {a, 1, nn}, {d, 0, nn}];
Sort[t, p[#1[[1]], #1[[2]]] < p[#2[[1]], #2[[2]]] &] // Flatten

Formula

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.

Extensions

a(37)-a(52) from Giovanni Resta, Apr 06 2017