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).

2, 17, 73, 193, 1873, 20353, 20353, 79633, 2333017

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

Table of n, a(n) for n=1..9.

Example

a(1) = 2 because, for A003173(1) = 1, 2 = 1^2+A003173(1)*1^2.
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:
a(6) = 63^2+A003173(1)*128^2 = 79^2+A003173(2)*84^2 = 55^2+A003173(3)*76^2 = 65^2+A003173(4)*48^2 = 137^2+A003173(5)*12^2 = 97^2+A003173(6)*24^2, with Heegner numbers up to A003173(6)=19, and also:
a(7) = 119^2+A003173(7)*12^2, with Heegner number A003173(7)=43.
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
          = 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)
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)))}
vector(9, n, a(n)) \\ Andrew Howroyd, Nov 05 2020

Crossrefs

Cf. A003173.

Sequence in context: A155715 A307851 A054568 * A235471 A297411 A183175

Adjacent sequences: A338084 A338085 A338086 * A338088 A338089 A338090

Keyword

fini,full,nonn

Author

Marco Frigerio, Oct 09 2020

Status

approved