A282093 -id:A282093 - cp's OEIS frontend

A282095 Larger member of a coprime pair (x,y) which solves x^2 + y^2 = z^3 with positive x, y and z.

11, 46, 52, 117, 142, 198, 236, 286, 415, 488, 524, 549, 621, 666, 835, 873, 908, 970, 1001, 1199, 1388, 1432, 1692, 1757, 1962, 1964, 1971, 2035, 2041, 2366, 2392, 2630, 2655, 2681, 2702, 2815, 2826, 3195, 3421, 3544, 3664, 3715, 4048, 4070, 4097, 4356

Offset: 1

R. J. Mathar, Feb 06 2017

nonn

If x and y are coprime, so obviously are also (x,z) and (y,z).
The ordered values of the bases of the cubes, z, are a subsequence of (and conjecturally the same as) A008846.
For production purposes we advice to use the parametrized representations (see references).

			2^2 + 11^2 = 5^3, so 11 is in the sequence.
9^2 + 46^2 = 13^3, so 46 is in the sequence.
47^2 + 52^2 = 17^3, so 52 is in the sequence.
44^2 + 117^2 = 25^2, so 117 is in the sequence.

R. J. Mathar, Table of n, a(n) for n = 1..1172
Imin Chen, On the equation s^2+y^(2p)=alpha^3, Math. Comp. 77 (262) (2008) 1223-1227.
Sander R. Dahmen, A refined modular approach to the diophantine equation x^2+y^(2n)=z^3, arXiv:1002.0020 [math.NT] (2010).

Subsequence of A282093. Cf. A099533.

# slow version for demonstration only.
isA282095 := proc(y)
    local x,z3 ;
    for x from 1 to y do
        if igcd(x,y) = 1 then
            z3 := x^2+y^2 ;
            if isA000578(z3) then
                return true ;
            end if;
        end if;
    end do:
    return false ;
end proc:
for y from 1 do
    if isA282095(y) then
        printf("%d,\n",y) ;
    end if;
end do:

okQ[y_] := Module[{x, z3}, For[x=1, xJean-François Alcover, Dec 04 2017, after R. J. Mathar *)

{y: x^2 + y^2 = z^3; gcd(x,y) = 1; 1 <= x <= y; x, y, z in N}

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

Original entry on oeis.org

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

R. J. Mathar, Feb 06 2017

nonn

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.

			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.

Cf. A282093.

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:

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 *)

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

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

Equals A282093 union A000578.

cp's OEIS Frontend

A282095 Larger member of a coprime pair (x,y) which solves x^2 + y^2 = z^3 with positive x, y and z.

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