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Larger member of a pair (x,y) which solves x^2 + y^2 = z^3 for nonnegative x, y and z.
0

%I #27 Nov 03 2023 06:18:33

%S 0,1,2,8,10,11,16,26,27,30,39,46,52,54,64,68,80,88,100,110,117,120,

%T 125,128,130,142,145,170,198,205,208,216,222,236,240,250,270,286,297,

%U 310,312,322,343,350,366,368,371,377,406,414,415,416,432,455,481

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

%C Values y such that x^2 + y^2 = z^3 has a solution 0 <= x <= y with integer x, y and z.

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

%F Equals A282093 union A000578.

%e 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.

%p isA282094 := proc(y)

%p local x,z3 ;

%p for x from 0 to y do

%p z3 := x^2+y^2 ;

%p if isA000578(z3) then

%p return true ;

%p end if;

%p end do:

%p return false ;

%p end proc:

%p for y from 0 to 800 do

%p if isA282094(y) then

%p printf("%d,",y) ;

%p end if;

%p end do:

%t 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]]];

%t 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_ *)

%o (Python)

%o from sympy import factorint

%o def is_cube(n):

%o if n==0: return True

%o return all(i%3==0 for i in factorint(n).values())

%o def ok(n):

%o return any(is_cube(x**2 + n**2) for x in range(n + 1))

%o print([n for n in range(501) if ok(n)]) # _Indranil Ghosh_, Jun 30 2017

%o (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

%Y Cf. A282093.

%K nonn

%O 1,3

%A _R. J. Mathar_, Feb 06 2017