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 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
 2, 17, 73, 193, 1873, 20353, 20353, 79633, 2333017 (list; graph; refs; listen; history; text; internal format)
 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==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

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Last modified October 4 01:05 EDT 2023. Contains 365872 sequences. (Running on oeis4.)