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

%I A282094 #27 Nov 03 2023 06:18:33
%S A282094 0,1,2,8,10,11,16,26,27,30,39,46,52,54,64,68,80,88,100,110,117,120,
%T A282094 125,128,130,142,145,170,198,205,208,216,222,236,240,250,270,286,297,
%U A282094 310,312,322,343,350,366,368,371,377,406,414,415,416,432,455,481
%N A282094 Larger member of a pair (x,y) which solves x^2 + y^2 = z^3 for nonnegative x, y and z.
%C A282094 Values y such that x^2 + y^2 = z^3 has a solution 0 <= x <= y with integer x, y and z.
%C A282094 Differs from A282093 because solutions with x=0 are admitted; (x,y) = (0,t^3) solves the equation with z = t^2.
%F A282094 Equals A282093 union A000578.
%e A282094 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 A282094 isA282094 := proc(y)
%p A282094     local x,z3 ;
%p A282094     for x from 0 to y do
%p A282094         z3 := x^2+y^2 ;
%p A282094         if isA000578(z3) then
%p A282094             return true ;
%p A282094         end if;
%p A282094     end do:
%p A282094     return false ;
%p A282094 end proc:
%p A282094 for y from 0 to 800 do
%p A282094     if isA282094(y) then
%p A282094         printf("%d,",y) ;
%p A282094     end if;
%p A282094 end do:
%t A282094 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 A282094 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 A282094 (Python)
%o A282094 from sympy import factorint
%o A282094 def is_cube(n):
%o A282094     if n==0: return True
%o A282094     return all(i%3==0 for i in factorint(n).values())
%o A282094 def ok(n):
%o A282094     return any(is_cube(x**2 + n**2) for x in range(n + 1))
%o A282094 print([n for n in range(501) if ok(n)]) # _Indranil Ghosh_, Jun 30 2017
%o A282094 (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 A282094 Cf. A282093.
%K A282094 nonn
%O A282094 1,3
%A A282094 _R. J. Mathar_, Feb 06 2017