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A124984 Primes of the form 8*k + 3 generated recursively. Initial prime is 3. General term is a(n) = Min_{p is prime; p divides 2 + Q^2; p == 3 (mod 8)}, where Q is the product of previous terms in the sequence.

%I A124984 #23 Aug 05 2022 10:50:39
%S A124984 3,11,1091,1296216011,2177870960662059587828905091,76870667,19,
%T A124984 257680660619,73677606898727076965233531,
%U A124984 23842300525435506904690028531941969449780447746432390747,35164737203
%N A124984 Primes of the form 8*k + 3 generated recursively. Initial prime is 3. General term is a(n) = Min_{p is prime; p divides 2 + Q^2; p == 3 (mod 8)}, where Q is the product of previous terms in the sequence.
%C A124984 2+Q^2 always has a prime divisor congruent to 3 modulo 8.
%D A124984 D. M. Burton, Elementary Number Theory, McGraw-Hill, Sixth Edition (2007), p. 191.
%H A124984 Robert Price, <a href="/A124984/b124984.txt">Table of n, a(n) for n = 1..15</a>
%H A124984 N. Hobson, <a href="http://www.qbyte.org/puzzles/">Home page (listed in lieu of email address)</a>
%e A124984 a(3) = 1091 is the smallest prime divisor congruent to 3 mod 8 of 2+Q^2 = 1091, where Q = 3 * 11.
%t A124984 a = {3}; q = 1;
%t A124984 For[n = 2, n ≤ 5, n++,
%t A124984     q = q*Last[a];
%t A124984     AppendTo[a, Min[Select[FactorInteger[2 + q^2][[All, 1]], Mod[#,
%t A124984     8] \[Equal] 3 &]]];
%t A124984     ];
%t A124984 a (* _Robert Price_, Jul 14 2015 *)
%o A124984 (PARI) lista(nn) = my(f, q=3); print1(q); for(n=2, nn, f=factor(2+q^2)[, 1]~; for(i=1, #f, if(f[i]%8==3, print1(", ", f[i]); q*=f[i]; break))); \\ _Jinyuan Wang_, Aug 05 2022
%Y A124984 Cf. A000945, A007520, A057204-A057208, A051308-A051335, A124984-A124993, A125037-A125045.
%K A124984 nonn
%O A124984 1,1
%A A124984 _Nick Hobson_, Nov 18 2006
%E A124984 a(10) from _Robert Price_, Jul 04 2015
%E A124984 a(11) from _Robert Price_, Jul 05 2015