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

1,1

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.

Links

Table of n, a(n) for n=1..20.

Example

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

Prog

(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(p<m, p=nextprime(p+1); if(isok(p, u), return(p))); return(0)}
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); }

Crossrefs

Cf. A000926, A003173, A340055, A340132.

Sequence in context: A374023 A198298 A074773 * A309003 A348811 A251523

Adjacent sequences: A340130 A340131 A340132 * A340134 A340135 A340136

Keyword

nonn

Author

Marco Frigerio, Dec 29 2020

Status

approved