A213158 Positive integers of the form (x+y+z)*x*y*z (x,y,z positive integers).

3, 8, 15, 20, 24, 35, 36, 48, 56, 63, 80, 84, 96, 99, 108, 120, 128, 135, 140, 143, 144, 168, 176, 180, 195, 200, 216, 224, 231, 240, 243, 255, 260, 264, 275, 288, 300, 308, 320, 323, 336, 351, 360, 384, 396, 399, 416, 420, 440, 455, 468, 476, 483, 495, 504

Offset: 1

Author

Jean-François Alcover, Jun 06 2012

Keywords

• nonn

Comments

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).

Positive integers of the form (a^2-b^2)*(b^2-c^2) with integers a>b>c>=0. - Michael Somos, May 18 2013

Examples

			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.
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
		

References

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

Crossrefs

Cf. A188158.

Programs

Mathematica

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]]]