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

%I A282095 #17 Dec 04 2017 03:01:48
%S A282095 11,46,52,117,142,198,236,286,415,488,524,549,621,666,835,873,908,970,
%T A282095 1001,1199,1388,1432,1692,1757,1962,1964,1971,2035,2041,2366,2392,
%U A282095 2630,2655,2681,2702,2815,2826,3195,3421,3544,3664,3715,4048,4070,4097,4356
%N A282095 Larger member of a coprime pair (x,y) which solves x^2 + y^2 = z^3 with positive x, y and z.
%C A282095 If x and y are coprime, so obviously are also (x,z) and (y,z).
%C A282095 The ordered values of the bases of the cubes, z, are a subsequence of (and conjecturally the same as) A008846.
%C A282095 For production purposes we advice to use the parametrized representations (see references).
%H A282095 R. J. Mathar, <a href="/A282095/b282095.txt">Table of n, a(n) for n = 1..1172</a>
%H A282095 Imin Chen, <a href="http://dx.doi.org/10.1090/S0025-5718-07-02083-2">On the equation s^2+y^(2p)=alpha^3</a>, Math. Comp. 77 (262) (2008) 1223-1227.
%H A282095 Sander R. Dahmen, <a href="http://arxiv.org/abs/1002.0020">A refined modular approach to the diophantine equation x^2+y^(2n)=z^3</a>, arXiv:1002.0020 [math.NT] (2010).
%F A282095 {y: x^2 + y^2 = z^3; gcd(x,y) = 1; 1 <= x <= y; x, y, z in N}
%e A282095 2^2 + 11^2 = 5^3, so 11 is in the sequence.
%e A282095 9^2 + 46^2 = 13^3, so 46 is in the sequence.
%e A282095 47^2 + 52^2 = 17^3, so 52 is in the sequence.
%e A282095 44^2 + 117^2 = 25^2, so 117 is in the sequence.
%p A282095 # slow version for demonstration only.
%p A282095 isA282095 := proc(y)
%p A282095     local x,z3 ;
%p A282095     for x from 1 to y do
%p A282095         if igcd(x,y) = 1 then
%p A282095             z3 := x^2+y^2 ;
%p A282095             if isA000578(z3) then
%p A282095                 return true ;
%p A282095             end if;
%p A282095         end if;
%p A282095     end do:
%p A282095     return false ;
%p A282095 end proc:
%p A282095 for y from 1 do
%p A282095     if isA282095(y) then
%p A282095         printf("%d,\n",y) ;
%p A282095     end if;
%p A282095 end do:
%t A282095 okQ[y_] := Module[{x, z3}, For[x=1, x<y, x++, If[CoprimeQ[x, y], z3 = x^2 + y^2; If[IntegerQ[z3^(1/3)], Print[y]; Return[True]]]]; Return[False]];
%t A282095 Select[Range[5000], okQ] (* _Jean-François Alcover_, Dec 04 2017, after _R. J. Mathar_ *)
%Y A282095 Subsequence of A282093. Cf. A099533.
%K A282095 nonn
%O A282095 1,1
%A A282095 _R. J. Mathar_, Feb 06 2017