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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. 0
3230498881, 5086789009, 6956459689, 7260636769, 12387462649, 13125124321, 14049841129, 14247509329, 14310889849, 15871864849, 16573389361, 17502040609, 17768627809, 22042168201, 22621870441, 22957650769, 23018043409, 23819076121, 25228204849, 26585136601 (list; graph; refs; listen; history; text; internal format)



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.


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


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


(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); }


Cf. A000926, A003173, A340055, A340132.

Sequence in context: A199630 A198298 A074773 * A309003 A251523 A155961

Adjacent sequences:  A340130 A340131 A340132 * A340134 A340135 A340136




Marco Frigerio, Dec 29 2020



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Last modified July 30 04:51 EDT 2021. Contains 346348 sequences. (Running on oeis4.)