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

Author

Marco Frigerio, Oct 09 2020

Keywords

• fini
• full
• nonn

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.

Examples

			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.
		

Crossrefs

Cf. A003173.

Programs

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