A340132 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).

1083289, 3818929, 6104641, 6868801, 7623529, 8465209, 9033649, 10105489, 11400481, 11597569, 11809561, 12338041, 12348961, 13154761, 13426009, 15861169, 16889161, 16922161, 18596449, 19684729, 20322481, 21067201, 21480001, 22684561, 23654569, 24531049

Offset

1,1

Comments

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

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.

Links

Table of n, a(n) for n=1..26.

Example

1083289 =  315^2  + A000926(1)*992^2
        = 1033^2  + A000926(2)*90^2
        =  979^2  + A000926(3)*204^2
        = ...
        =  817^2  + A000926(65)*15^2.

Prog

(PARI) Idoneal()={return(select(m->!#select(k->k<>2, quadclassunit(-4*m).cyc), [1..1848])); }
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=Idoneal()); while(p<m, p=nextprime(p+1); if(isok(p, u), return(p))); return(0)}
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); }

Crossrefs

Cf. A000926, A338088.

Sequence in context: A236449 A352110 A253467 * A340405 A258704 A224592

Adjacent sequences: A340129 A340130 A340131 * A340133 A340134 A340135

Keyword

nonn

Author

Marco Frigerio, Dec 29 2020

Status

approved