cp's OEIS Frontend

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

%I A099426 #25 Jan 22 2025 05:36:06
%S A099426 3,228,671,1261,6371,9765,35113,35928,40380,41643,66599,112245,124501,
%T A099426 127499,167160,191771,205485,255720,297037,377567,532392,546013,
%U A099426 647569,681285,812340,897623,1043469,1125683,1261491,1431793,1433040,1584828,1783067,1984009,2107391,2372903,2440893,2484469,2548557
%N A099426 Numbers n where n^2 = x^3 + y^3; x,y>0 and gcd(x,y)=1.
%C A099426 Based on an observation of _Ed Pegg Jr_, who supplied terms a(2)-a(6) and a(8).
%H A099426 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 A099426 228 is in the sequence because 228^2 = 11^3 + 37^3 and gcd(11, 37) = 1.
%t A099426 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 A099426 (PARI)
%o A099426 is_A099426(n)=
%o A099426 {
%o A099426     my(n2=n^2, k=1, k3=1, r);
%o A099426     while( k3 < n2,
%o A099426         if ( ispower(n2-k3, 3, &r),
%o A099426             if ( gcd(r,k)==1, return(1) );
%o A099426         );
%o A099426         k+=1;  k3=k^3;
%o A099426     );
%o A099426     return(0);
%o A099426 }
%o A099426 for (n=1,10^8, if( is_A099426(n), print1(n,", ")) );
%o A099426 /* _Joerg Arndt_, Sep 30 2012 */
%o A099426 (PARI)
%o A099426 /* compute all terms below a threshold at once, terms need to be sorted */
%o A099426 { N = 10^7; N2 = N^2;
%o A099426 x=1; x3=x^3;
%o A099426 while ( x3 < N2,
%o A099426     y=x+1; y3=y^3;
%o A099426     while ( y3 < N2,
%o A099426         if ( gcd(x,y) == 1,
%o A099426             s = x3 + y3;
%o A099426             if ( issquare(s, &r), print(r); break(); );
%o A099426         );
%o A099426         y+=1;  y3 = y^3;
%o A099426     );
%o A099426     x+=1;  x3 = x^3;
%o A099426 );}
%o A099426 /* _Joerg Arndt_, Sep 30 2012 */
%o A099426 (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 A099426 Cf. A099532, A099533, A103255 (min(x,y), sorted), A282639 (max(x,y)).
%K A099426 nonn
%O A099426 1,1
%A A099426 _Hans Havermann_, Oct 15 2004
%E A099426 More terms from _Hans Havermann_ and _Bodo Zinser_, Oct 20 2004