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A057204 Primes congruent to 1 mod 6 generated recursively. Initial prime is 7. The next term is p(n) = Min_{p is prime; p divides 4Q^2+3; p mod 6 = 1}, where Q is the product of previous entries of the sequence.

%I A057204 #27 Dec 12 2021 20:00:15
%S A057204 7,199,7761799,487,67,103,3562539697,7251847,13,127,
%T A057204 5115369871402405003,31,697830431171707,151,3061,229,193,
%U A057204 5393552285540920774057256555028583857599359699,709,397,37,61,46168741,3127279,181,122268541
%N A057204 Primes congruent to 1 mod 6 generated recursively. Initial prime is 7. The next term is p(n) = Min_{p is prime; p divides 4Q^2+3; p mod 6 = 1}, where Q is the product of previous entries of the sequence.
%C A057204 4*Q^2 + 3 always has a prime divisor congruent to 1 modulo 6.
%C A057204 If we start with the empty product Q=1 then it is not necessary to specify the initial prime. - _Jens Kruse Andersen_, Jun 30 2014
%D A057204 P. G. L. Dirichlet (1871): Vorlesungen uber Zahlentheorie. Braunschweig, Viewig, Supplement VI, 24 pages.
%D A057204 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, page 13.
%H A057204 Sean A. Irvine, <a href="/A057204/b057204.txt">Table of n, a(n) for n = 1..48</a>
%e A057204 a(4)=487 is the smallest prime divisor of 4*Q*Q + 3 = 10812186007, congruent to 1 (mod 6), where Q = 7*199*7761799.
%t A057204 a={7}; q=1;
%t A057204 For[n=2,n<=7,n++,
%t A057204     q=q*Last[a];
%t A057204     AppendTo[a,Min[Select[FactorInteger[4*q^2+3][[All,1]],Mod[#,6]==1 &]]];
%t A057204     ];
%t A057204 a (* _Robert Price_, Jul 16 2015 *)
%o A057204 (PARI) Q=1;for(n=1,11,f=factor(4*Q^2+3);for(i=1,#f~,p=f[i,1];if(p%6==1,break));print1(p", ");Q*=p) \\ _Jens Kruse Andersen_, Jun 30 2014
%Y A057204 Cf. A000945, A000946, A005265, A005266, A051308-A051335, A002476, A057204-A057208.
%K A057204 nonn
%O A057204 1,1
%A A057204 _Labos Elemer_, Oct 09 2000
%E A057204 More terms from Nick Hobson, Nov 14 2006
%E A057204 More terms from _Sean A. Irvine_, Oct 23 2014