A218467 - OEIS

%I #21 Jul 16 2015 13:59:14

%S 2,3,7,43,139,50207,23,10651,563,11,19,363303615453958067659,787,2803,

%T 3261639461817858097484047657974700766171,

%U 448513341328399688966874038187266281752082128599801650127,89724193529143

%N A variant of the Euclid-Mullin sequence A000945: a(1) = 2, a(n+1) is smallest prime factor congruent to 3 (mod 4) of Product_{k=1..n} a(k) + 1.

%C Just as the Euclid-Mullin sequence is suggested by Euclid's proof of an infinity of primes, this sequence is suggested by a variation of his proof, showing the existence of an infinity of primes congruent to 3 (mod 4).  See Hardy and Wright in the Reference below.

%C Could also be viewed as a variation on A217759.  Restricting the scope of "smallest prime factor congruent to 3 (mod 4)" to the larger of the two algebraic factors of 4Q^2-1 as defined in that sequence results in a sequence essentially the same as this one.

%C a(18) has 149 digits.

%D P. G. L. Dirichlet: Vorlesungen uber Zahlentheorie. Braunschweig, Viewig, Supplement VI, (1871), 24 pages.

%D G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 13.

%H Daran Gill, <a href="/A218467/b218467.txt">Table of n, a(n) for n = 1..19</a>

%e This sequence and A000945 are identical up to their fourth term.  The fifth terms of both that sequence and this one are factors of 2*3*7*43+1=13*139.  The smallest factor, used by A000945, is congruent to 1 (mod 4).  Here we take the larger.

%t a={2}; q=1;

%t For[n=2,n<=12,n++,

%t     q=q*Last[a];

%t     AppendTo[a,Min[Select[FactorInteger[q+1][[All,1]],Mod[#,4]==3 &]]];

%t     ];

%t a (* _Robert Price_, Jul 16 2015 *)

%K nonn

%O 1,1

%A _Daran Gill_, Mar 26 2013