A282095 - OEIS

%I #17 Dec 04 2017 03:01:48

%S 11,46,52,117,142,198,236,286,415,488,524,549,621,666,835,873,908,970,

%T 1001,1199,1388,1432,1692,1757,1962,1964,1971,2035,2041,2366,2392,

%U 2630,2655,2681,2702,2815,2826,3195,3421,3544,3664,3715,4048,4070,4097,4356

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

%C If x and y are coprime, so obviously are also (x,z) and (y,z).

%C The ordered values of the bases of the cubes, z, are a subsequence of (and conjecturally the same as) A008846.

%C For production purposes we advice to use the parametrized representations (see references).

%H R. J. Mathar, <a href="/A282095/b282095.txt">Table of n, a(n) for n = 1..1172</a>

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

%e 2^2 + 11^2 = 5^3, so 11 is in the sequence.

%e 9^2 + 46^2 = 13^3, so 46 is in the sequence.

%e 47^2 + 52^2 = 17^3, so 52 is in the sequence.

%e 44^2 + 117^2 = 25^2, so 117 is in the sequence.

%p # slow version for demonstration only.

%p isA282095 := proc(y)

%p     local x,z3 ;

%p     for x from 1 to y do

%p         if igcd(x,y) = 1 then

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

%p             if isA000578(z3) then

%p                 return true ;

%p             end if;

%p         end if;

%p     end do:

%p     return false ;

%p end proc:

%p for y from 1 do

%p     if isA282095(y) then

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

%p     end if;

%p end do:

%t 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 Select[Range[5000], okQ] (* _Jean-François Alcover_, Dec 04 2017, after _R. J. Mathar_ *)

%Y Subsequence of A282093. Cf. A099533.

%K nonn

%O 1,1

%A _R. J. Mathar_, Feb 06 2017