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Positive integers of the form (x+y+z)*x*y*z (x,y,z positive integers).
2

%I #18 May 20 2013 02:52:44

%S 3,8,15,20,24,35,36,48,56,63,80,84,96,99,108,120,128,135,140,143,144,

%T 168,176,180,195,200,216,224,231,240,243,255,260,264,275,288,300,308,

%U 320,323,336,351,360,384,396,399,416,420,440,455,468,476,483,495,504

%N Positive integers of the form (x+y+z)*x*y*z (x,y,z positive integers).

%C Square terms are 36, 144, 576,... and the corresponding square roots are 6, 12, 24,... i.e. sequence A188158 (integer areas of primitive integer triangles).

%C Positive integers of the form (a^2-b^2)*(b^2-c^2) with integers a>b>c>=0. - _Michael Somos_, May 18 2013

%D R. D. Carmichael, Diophantine Analysis, Wiley, 1915, p. 9.

%e a(21)=144 for x=1, y=4 and z=4 then the triangle sides are x+y = 5, z+x = 5 and y+z = 8, hence half-perimeter = p = x+y+z = 9 and Heron's formula is checked: area = sqrt(p*(p-5)*(p-5)*(p-8)) = sqrt(144) = 12.

%e 36 = (4^2-2^2) * (2^2-1^2). 63 = (5^2-2^2) * (2^2-1^2) = (5^2-4^2) * (4^2-2^2)= (8^2-1^2) * (1^2-0^2). - _Michael Somos_, May 19 2013

%t nmax = 25; mx = nmax (nmax + 2); Union[Reap[Do[a = (x + y + z)*x*y*z; If[a <= mx, Sow[a]], {x, 1, nmax}, {y, x, nmax}, {z, y, nmax}]][[2, 1]]]

%Y Cf. A188158.

%K nonn

%O 1,1

%A _Jean-François Alcover_, Jun 06 2012