A225870 Nonnegative integers of the form x*y*z*(x+y-z) with integers x>=y>=z.

0, 1, 4, 9, 12, 16, 24, 25, 36, 40, 45, 49, 60, 64, 72, 81, 84, 100, 105, 112, 120, 121, 144, 160, 169, 180, 189, 192, 196, 216, 220, 225, 240, 252, 256, 264, 280, 289, 297, 300, 312, 324, 336, 352, 360, 361, 364, 384, 385, 396, 400, 420, 429, 432, 441, 480

Offset: 1

Author

Michael Somos, May 18 2013

Keywords

• nonn

Comments

For n>=0 and n = x*y*z*(x+y-z) with integers x>=y>=z then we can even find nonnegative solutions (x,y,z). However, if we restrict to z>=0 then there are no solutions (x,y,z) in case n<0.

The negative integers of the form x*y*z*(x+y-z) with integers x>=y>=z are the negatives of A213158 and in that case z<0.

Nonnegative integers of the form (a^2-c^2)*(b^2-c^2) with integers a>=b>=c.

Note that we must allow c<0 to represent n=12, 24, 40, ....

The negative integers of the form (a^2-c^2)*(b^2-c^2) with integers a>=b>=c are the negatives of A213158.

Examples

			12 = (1)*(-2)*(-3)*((1)+(-2)-(-3)) with (x,y,z) = (1,-2,-3).
12 = 2*2*1*(2+2-1) with (x,y,z) = (2,2,1).
12 = ((0)^2-(-2)^2)*((-1)^2-(-2)^2) with (a,b,c) = (0,-1,-2).
12 = ((1)^2-(-2)^2)*((0)^2-(-2)^2) with (a,b,c) = (1,0,-2).
		

Crossrefs

Cf. A213158.

Programs

PARI

{isa(n) = forvec( v = vector(3, i, [0, ceil(n^(1/2))]), if( n == v[1] * v[2] * v[3] * (v[3] + v[2] - v[1]), return(1)), 1)}