A340132 - OEIS

%I #21 Apr 01 2021 23:09:54

%S 1083289,3818929,6104641,6868801,7623529,8465209,9033649,10105489,

%T 11400481,11597569,11809561,12338041,12348961,13154761,13426009,

%U 15861169,16889161,16922161,18596449,19684729,20322481,21067201,21480001,22684561,23654569,24531049

%N Least prime numbers, in ascending order, such that each of them can be written, in a unique way, in the form x^2 + h*y^2, where x, y are natural numbers, while h takes all the values of the sequence A000926 (idoneal numbers).

%C First number in this sequence is equal to last number of sequence A338088.

%C The sequence is obtained using Lista(m), with m=246*10^5, see section PROG. It's possible to increase m to discover more terms of the sequence.

%e 1083289 =  315^2  + A000926(1)*992^2

%e         = 1033^2  + A000926(2)*90^2

%e         =  979^2  + A000926(3)*204^2

%e         = ...

%e         =  817^2  + A000926(65)*15^2.

%o (PARI) Idoneal()={return(select(m->!#select(k->k<>2, quadclassunit(-4*m).cyc), [1..1848]));}

%o isok(p,u)={my (i, s, n=matsize(u)[2], t=0);for(i=1, n, s=kronecker(-u[i],p); if(s==1, t++,break));if(t==n,t=0;for(i=1, n, s=qfbsolve(Qfb(1,0,u[i]),p); if(s==[], break,t++)));if(t==n,1,0)}

%o Primo(p, m)={my(u=Idoneal()); while(p<m, p=nextprime(p+1); if(isok(p,u),return(p)));return(0)}

%o Lista(m)={ my (q,r=108*10^4,v=[]); q=nextprime(r); m=precprime(m); while(q<m,r=q;q=Primo(r,m);if(q>r,v=concat(v,q),q=m)); return(v);}

%Y Cf. A000926, A338088.

%K nonn

%O 1,1

%A _Marco Frigerio_, Dec 29 2020