%I #11 Mar 25 2018 08:26:00

%S 1,1,2,3,4,6,9,12,14,16,19,24,25,26

%N Records of the maximum number of distinct rational solutions to x^2 - Dy^2 = t, 0 < D <= k, 0 < t <= k.

%C Values of k are in A240469.

%e All Diophantine equations x^2 - Dy^2 = t, 0 < D <= 16, 0 < t <= 16, D squarefree, have fewer than 4 distinct solutions; the first with 4 solutions is x^2 - 17y^2 = 16 with the solutions (x,y) = (9/2,1/2), (21,5), (4,0), (13,3), so 4 is in sequence.

%o (PARI) { r(l,k)=if(!issquarefree(l)||!polisirreducible(z^2-l),return(0));v=bnfisintnorm(bnfinit(z^2-l), k);if(!#v,return(0));s=0;for(k=1,#v,p=v[k];a=polcoeff(p,0);b=polcoeff(p,1);f=1;for(l=k+1,#v,p=v[l];aa=polcoeff(p,0);bb=polcoeff(p,1);if(abs(a)==abs(aa)&&abs(b)==abs(bb),f=0;break));s=s+f);s

%o m=0;n=0;while(1,n=n+1;res=0;for(l=1,n,rr=r(l,n);if(rr>res,res=rr));for(k=1,n-1,rr=r(n,k);if(rr>res,res=rr));if(res>m,m=res;print(res,","))) }

%K nonn,more

%O 1,3

%A _Ralf Stephan_, Apr 06 2014