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A057207 a(1)=5, a(n) is the smallest prime dividing 4*Q^2 + 1 where Q is the product of all previous terms in the sequence.

%I A057207 #28 Aug 27 2014 10:42:19
%S A057207 5,101,1020101,53,29,2507707213238852620996901,449,13,8693,1997,6029,
%T A057207 61,3181837,113,181,1934689,
%U A057207 6143090225314378441493352126119201470973493456817556328833988172277,4733,3617,41,68141,37,51473,17,821,598201519454797,157,9689,2357,757,149,293,5261
%N A057207 a(1)=5, a(n) is the smallest prime dividing 4*Q^2 + 1 where Q is the product of all previous terms in the sequence.
%C A057207 Removed redundant mod(p,4) = 1 criterion from definition.  By quadratic reciprocity, all factors of 1 + 4Q^2 are congruent to 1 (mod 4). See comments at the end of the b-file for an additional eight terms not proved, but nevertheless highly likely to be correct. - _Daran Gill_, Mar 23 2013
%D A057207 P. G. L. Dirichlet (1871): Vorlesungen über Zahlentheorie. Braunschweig, Viewig, Supplement VI, 24 pages.
%D A057207 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, page 13.
%H A057207 Daran Gill, <a href="/A057207/b057207.txt">Table of n, a(n) for n = 1..33</a>
%H A057207 Mersenne Forum, <a href="http://mersenneforum.org/showthread.php?t=17990">Sequence A057207</a>
%H A057207 OEIS wiki, <a href="https://oeis.org/wiki/OEIS_sequences_needing_factors">OEIS sequences needing factors</a>
%e A057207 a(4)=53 is the smallest prime divisor of 4*(5.101.1020101)^2+1 = 1061522231810040101 = 53*1613*12417062216309.
%t A057207 t = {5}; Do[q = Times @@ t; AppendTo[t, FactorInteger[1 + 4*q^2][[1, 1]]], {6}]; t (* _T. D. Noe_, Mar 27 2013 *)
%Y A057207 Cf. A000945, A000946, A005265, A005266, A051308-A051335, A002144, A057204-A057208.
%K A057207 nonn
%O A057207 1,1
%A A057207 _Labos Elemer_, Oct 09 2000
%E A057207 Eight more terms, a(9)-a(16), from _Max Alekseyev_, Apr 27 2009
%E A057207 Seventeen more terms, a(17)-a(33), added by _Daran Gill_, Mar 23 2013