A282093 Larger member of a pair (x,y) which solves x^2+y^2 = z^3 for positive x, y and z.

2, 10, 11, 16, 26, 30, 39, 46, 52, 54, 68, 80, 88, 100, 110, 117, 120, 128, 130, 142, 145, 170, 198, 205, 208, 222, 236, 240, 250, 270, 286, 297, 310, 312, 322, 350, 366, 368, 371, 377, 406, 414, 415, 416, 432, 455, 481, 488, 505, 518, 520, 524, 544, 549, 584

Offset: 1

Author

R. J. Mathar, Feb 06 2017

Keywords

• nonn

Comments

Values y such that x^2+y^2 = z^3 has a solution 1<=x<=y with integer x, y and z.

The positive values of A033431 are a subsequence, induced by solutions where x=y.

There are entries which have more than one representation, e.g., 10^2 + 198^2 = 34^3 and 107^2 + 198^2 = 37^3 both with y=198. 234^2 + 415^2 = 61^3 and 320^2 + 415^2 = 65^3 both with y=415.

The ordered sequence of x can apparently be constructed by retrieving the perfect squares in A106265 and printing their square roots: 1, 2, 5, 7, 8, 9, 10, 11, 16, 17, 18 , 26, 27, 30,...

Examples

			2^2+2^2=2^3, so 2 is in. 5^2+10^2=5^3, so 10 is in. 2^2+11^2 = 5^3, so 11 is in. 16^2+16^2=8^3, so 16 is in.
		

Crossrefs

Cf. A000404 (values of z), A033431, A106265.

Programs

Maple

isA282093 := proc(y)
    local x,z3 ;
    for x from 1 to y do
        z3 := x^2+y^2 ;
        if isA000578(z3) then
            return true ;
        end if;
    end do:
    return false ;
end proc:
for y from 1 to 800 do
    if isA282093(y) then
        printf("%d,\n",y) ;
    end if;
end do:

Mathematica

isA282093[y_] := Module[{x, z3},
For[x = 1, x <= y, x++, z3 = x^2+y^2; If[IntegerQ[z3^(1/3)], Return[True]]]; Return[False]];
Reap[For[y = 1, y <= 800, y++, If[isA282093[y], Print[y]; Sow[y]]]][[2, 1]] (* Jean-François Alcover, May 29 2023, after R. J. Mathar *)