

A338087


a(n) is the smallest prime number which can be represented as x^2 + h*y^2 with x > 0 and y > 0 for each h in the first n Heegner numbers (A003173).


1




OFFSET

1,1


COMMENTS

The sequence lists prime numbers, in nondecreasing 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 an increasing number of values of the sequence A003173 (Heegner numbers). See examples.


LINKS



EXAMPLE

a(2) = 17 because, considered the first two Heegner numbers, A003173(1) = 1 and A003173(2) = 2, 17 = 1^2+A003173(1)*4^2 = 3^2+A003173(2)*2^2.
The prime 20353 is present in the sequence 2 times because:
2333017 is the last term of the sequence since for every Heegner number h there are x, y such that 2333017 = x^2 + h*y^2 and this is the least prime for which this is possible.
For n=9, h in A003173 = {1,2,3,7,11,19,43,67,163},
a(9) = 2333017


PROG

(PARI)
isok(p, u)={for(i=1, #u, my(s=qfbsolve(Qfb(1, 0, u[i]), p)); if(s==0  s[1]==0, return(0))); 1}
a(n)={my(u=[1, 2, 3, 7, 11, 19, 43, 67, 163][1..n]); forprime(p=2, oo, if(isok(p, u), return(p)))}


CROSSREFS



KEYWORD

fini,full,nonn


AUTHOR



STATUS

approved



