A340055 Primes that can be written in the form j^2 + h*k^2, where j and k are positive integers, for every h in A003173 (Heegner numbers).

2333017, 5995081, 11414209, 11941273, 12953593, 14823769, 18550849, 19231969, 23582161, 26603977, 27336457, 29236729, 32630161, 35452033, 35836249, 37895089, 40411177, 42911257, 46007329, 46087057, 49680577, 49825609, 52046593, 52208017, 55624297, 63257401

Offset

1,1

Comments

The first term in this sequence is equal to last term in A338087.

The sequence is obtained using Lista(m), with m=633*10^5, see section PROG. One can increase m to obtain further terms of the sequence.

Links

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

Example

2333017 =  989^2 + A003173(1)*1164^2
        = 1493^2 + A003173(2)*228^2
        = 1093^2 + A003173(3)*616^2
        =  685^2 + A003173(4)*516^2
        = 1349^2 + A003173(5)*216^2
        =  179^2 + A003173(6)*348^2
        = 1293^2 + A003173(7)*124^2
        = 1395^2 + A003173(8)*76^2
        = 1485^2 + A003173(9)*28^2.

Prog

(PARI) Heegner()={my (d, k, v);  v=vector(3, i, i); for(k=2, 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=Heegner()); while(p<m, p=nextprime(p+1); if(isok(p, u), return(p))); return(0)}
Lista(m)={my (q, r=233*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. A003173, A338087.

Sequence in context: A154676 A250926 A278200 * A246226 A204944 A184771

Adjacent sequences: A340052 A340053 A340054 * A340056 A340057 A340058

Keyword

nonn

Author

Marco Frigerio, Dec 29 2020

Status

approved