A099426 Numbers n where n^2 = x^3 + y^3; x,y>0 and gcd(x,y)=1.

3, 228, 671, 1261, 6371, 9765, 35113, 35928, 40380, 41643, 66599, 112245, 124501, 127499, 167160, 191771, 205485, 255720, 297037, 377567, 532392, 546013, 647569, 681285, 812340, 897623, 1043469, 1125683, 1261491, 1431793, 1433040, 1584828, 1783067, 1984009, 2107391, 2372903, 2440893, 2484469, 2548557

Offset: 1

Author

Hans Havermann, Oct 15 2004

Keywords

• nonn

Comments

Based on an observation of Ed Pegg Jr, who supplied terms a(2)-a(6) and a(8).

Examples

			228 is in the sequence because 228^2 = 11^3 + 37^3 and gcd(11, 37) = 1.
		

Links

• Joerg Arndt and Donovan Johnson, Table of n, a(n) for n = 1..300 (first 55 terms from Joerg Arndt)

Crossrefs

Cf. A099532, A099533, A103255 (min(x,y), sorted), A282639 (max(x,y)).

Programs

Mathematica

n = 10^7; n2 = n^2; x = 1; x3 = x^3; Reap[ While[x3 < n2, y = x + 1; y3 = y^3; While[y3 < n2, If[GCD[x, y] == 1, s = x3 + y3; If[ IntegerQ[r = Sqrt[s]], Print[r]; Sow[r]; Break[]]]; y += 1; y3 = y^3]; x += 1; x3 = x^3]][[2, 1]] // Sort (* Jean-François Alcover, Jan 11 2013, translated from Joerg Arndt's 2nd Pari program *)

PARI

is_A099426(n)=
{
    my(n2=n^2, k=1, k3=1, r);
    while( k3 < n2,
        if ( ispower(n2-k3, 3, &r),
            if ( gcd(r,k)==1, return(1) );
        );
        k+=1;  k3=k^3;
    );
    return(0);
}
for (n=1,10^8, if( is_A099426(n), print1(n,", ")) );
/* Joerg Arndt, Sep 30 2012 */

PARI

/* compute all terms below a threshold at once, terms need to be sorted */
{ N = 10^7; N2 = N^2;
x=1; x3=x^3;
while ( x3 < N2,
    y=x+1; y3=y^3;
    while ( y3 < N2,
        if ( gcd(x,y) == 1,
            s = x3 + y3;
            if ( issquare(s, &r), print(r); break(); );
        );
        y+=1;  y3 = y^3;
    );
    x+=1;  x3 = x^3;
);}
/* Joerg Arndt, Sep 30 2012 */

PARI

for(s=2,1e5,for(x=1,s\2,my(y=s-x);if(gcd(x,y)>1,next); if(issquare(x^3+y^3), print1(s", ")))) \\ Charles R Greathouse IV, Nov 06 2014

Extensions

More terms from Hans Havermann and Bodo Zinser, Oct 20 2004