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A099426 Numbers n where n^2 = x^3 + y^3; x,y>0 and gcd(x,y)=1. 7

%I

%S 3,228,671,1261,6371,9765,35113,35928,40380,41643,66599,112245,124501,

%T 127499,167160,191771,205485,255720,297037,377567,532392,546013,

%U 647569,681285,812340,897623,1043469,1125683,1261491,1431793,1433040,1584828,1783067,1984009,2107391,2372903,2440893,2484469,2548557

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

%C Based on an observation of _Ed Pegg Jr_, who supplied terms a(2)-a(6) and a(8).

%H Joerg Arndt and Donovan Johnson, <a href="/A099426/b099426.txt">Table of n, a(n) for n = 1..300</a> (first 55 terms from Joerg Arndt)

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

%t 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 *)

%o (PARI)

%o is_A099426(n)=

%o {

%o my(n2=n^2, k=1, k3=1, r);

%o while( k3 < n2,

%o if ( ispower(n2-k3, 3, &r),

%o if ( gcd(r,k)==1, return(1) );

%o );

%o k+=1; k3=k^3;

%o );

%o return(0);

%o }

%o for (n=1,10^8, if( is_A099426(n), print1(n,", ")) );

%o /* _Joerg Arndt_, Sep 30 2012 */

%o (PARI)

%o /* compute all terms below a threshold at once, terms need to be sorted */

%o { N = 10^7; N2 = N^2;

%o x=1; x3=x^3;

%o while ( x3 < N2,

%o y=x+1; y3=y^3;

%o while ( y3 < N2,

%o if ( gcd(x,y) == 1,

%o s = x3 + y3;

%o if ( issquare(s, &r), print(r); break(); );

%o );

%o y+=1; y3 = y^3;

%o );

%o x+=1; x3 = x^3;

%o );}

%o /* _Joerg Arndt_, Sep 30 2012 */

%o (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

%Y Cf. A099532, A099533, A103255 (min(x,y), sorted).

%K nonn

%O 1,1

%A _Hans Havermann_, Oct 15 2004

%E More terms from _Hans Havermann_ and _Bodo Zinser_, Oct 20 2004

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Last modified July 30 16:50 EDT 2021. Contains 346359 sequences. (Running on oeis4.)