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

%I A282093 #17 May 29 2023 11:32:06
%S A282093 2,10,11,16,26,30,39,46,52,54,68,80,88,100,110,117,120,128,130,142,
%T A282093 145,170,198,205,208,222,236,240,250,270,286,297,310,312,322,350,366,
%U A282093 368,371,377,406,414,415,416,432,455,481,488,505,518,520,524,544,549,584
%N A282093 Larger member of a pair (x,y) which solves x^2+y^2 = z^3 for positive x, y and z.
%C A282093 Values y such that x^2+y^2 = z^3 has a solution 1<=x<=y with integer x, y and z.
%C A282093 The positive values of A033431 are a subsequence, induced by solutions where x=y.
%C A282093 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.
%C A282093 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,...
%e A282093 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.
%p A282093 isA282093 := proc(y)
%p A282093     local x,z3 ;
%p A282093     for x from 1 to y do
%p A282093         z3 := x^2+y^2 ;
%p A282093         if isA000578(z3) then
%p A282093             return true ;
%p A282093         end if;
%p A282093     end do:
%p A282093     return false ;
%p A282093 end proc:
%p A282093 for y from 1 to 800 do
%p A282093     if isA282093(y) then
%p A282093         printf("%d,\n",y) ;
%p A282093     end if;
%p A282093 end do:
%t A282093 isA282093[y_] := Module[{x, z3},
%t A282093 For[x = 1, x <= y, x++, z3 = x^2+y^2; If[IntegerQ[z3^(1/3)], Return[True]]]; Return[False]];
%t A282093 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_ *)
%Y A282093 Cf. A000404 (values of z), A033431, A106265.
%K A282093 nonn
%O A282093 1,1
%A A282093 _R. J. Mathar_, Feb 06 2017