cp's OEIS Frontend

A340133 The sequence lists the 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 sequences A000926 (Idoneal numbers) and A003173 (Heegner numbers). See example.

%I A340133 #20 Apr 01 2021 23:10:15
%S A340133 3230498881,5086789009,6956459689,7260636769,12387462649,13125124321,
%T A340133 14049841129,14247509329,14310889849,15871864849,16573389361,
%U A340133 17502040609,17768627809,22042168201,22621870441,22957650769,23018043409,23819076121,25228204849,26585136601
%N A340133 The sequence lists the 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 sequences A000926 (Idoneal numbers) and A003173 (Heegner numbers). See example.
%C A340133 First number in this sequence is equal to least common number of sequences A340055 and A340132.
%C A340133 The sequence is obtained using Lista(m), with m=266*10^8, see section PROG. It's possible increase m to discover more terms of the sequence. It's also possible to extend the sequences A340055 and A340132 to check their common numbers.
%e A340133 3230498881 = 2465^2+A000926(1)*56784^2
%e A340133            = 56609^2+A000926(2)*3600^2
%e A340133            = 35927^2+A000926(3)*25428^2
%e A340133            = ...
%e A340133            = 56791^2+A003173(9)*180^2
%e A340133            = ...
%e A340133            = 35743^2+A000926(65)*1028^2
%o A340133 (PARI) Union()={ my (v);v=(select(m->!#select(k->k<>2, quadclassunit(-4*m).cyc), [1..1848]));for(k=3, 41, d=4*k-1; if(isprime(d) && qfbclassno(-d)==1, v=concat(v, d)));return(v);}
%o A340133 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 A340133 Primo(p, m)={my(u=Union()); while(p<m, p=nextprime(p+1); if(isok(p,u),return(p)));return(0)}
%o A340133 Lista(m)={ my (q,r=323*10^7,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 A340133 Cf. A000926, A003173, A340055, A340132.
%K A340133 nonn
%O A340133 1,1
%A A340133 _Marco Frigerio_, Dec 29 2020