A282094 Larger member of a pair (x,y) which solves x^2 + y^2 = z^3 for nonnegative x, y and z.

0, 1, 2, 8, 10, 11, 16, 26, 27, 30, 39, 46, 52, 54, 64, 68, 80, 88, 100, 110, 117, 120, 125, 128, 130, 142, 145, 170, 198, 205, 208, 216, 222, 236, 240, 250, 270, 286, 297, 310, 312, 322, 343, 350, 366, 368, 371, 377, 406, 414, 415, 416, 432, 455, 481

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 0 <= x <= y with integer x, y and z.

Differs from A282093 because solutions with x=0 are admitted; (x,y) = (0,t^3) solves the equation with z = t^2.

Examples

			0^2 + 0^2 = 0^3, so 0 is in. 0^2 + 1^2 = 1^3, so 1 is in. 2^2 + 2^2 = 2^3, so 2 is in. 0^2 + 8^2 = 4^3, so 8 is in. 5^2 + 10^2 = 5^3, so 10 is in.
		

Crossrefs

Cf. A282093.

Programs

Maple

isA282094 := proc(y)
    local x,z3 ;
    for x from 0 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 0 to 800 do
    if isA282094(y) then
        printf("%d,",y) ;
    end if;
end do:

Mathematica

isA282094[y_] := If[IntegerQ[y^(1/3)], True, 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 = 0, y <= 800, y++, If[isA282094[y], Print[y]; Sow[y]]]][[2, 1]] (* Jean-François Alcover, Nov 03 2023, after R. J. Mathar *)

PARI

is(n)=my(n2=n^2); for(x=0,n, if(ispower(n2+x^2,3), return(1))); 0 \\ Charles R Greathouse IV, Jun 30 2017

Python

from sympy import factorint
def is_cube(n):
    if n==0: return True
    return all(i%3==0 for i in factorint(n).values())
def ok(n):
    return any(is_cube(x**2 + n**2) for x in range(n + 1))
print([n for n in range(501) if ok(n)]) # Indranil Ghosh, Jun 30 2017

Formula

Equals A282093 union A000578.