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.

3230498881, 5086789009, 6956459689, 7260636769, 12387462649, 13125124321, 14049841129, 14247509329, 14310889849, 15871864849, 16573389361, 17502040609, 17768627809, 22042168201, 22621870441, 22957650769, 23018043409, 23819076121, 25228204849, 26585136601

Offset: 1

Author

Marco Frigerio, Dec 29 2020

Keywords

• nonn

Comments

First number in this sequence is equal to least common number of sequences A340055 and A340132.

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.

Examples

			3230498881 = 2465^2+A000926(1)*56784^2
           = 56609^2+A000926(2)*3600^2
           = 35927^2+A000926(3)*25428^2
           = ...
           = 56791^2+A003173(9)*180^2
           = ...
           = 35743^2+A000926(65)*1028^2
		

Crossrefs

Cf. A000926, A003173, A340055, A340132.

Programs

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);}
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)}
Primo(p, m)={my(u=Union()); while(pr,v=concat(v,q),q=m)); return(v);}