A218467 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.

2, 3, 7, 43, 139, 50207, 23, 10651, 563, 11, 19, 363303615453958067659, 787, 2803, 3261639461817858097484047657974700766171, 448513341328399688966874038187266281752082128599801650127, 89724193529143

Offset

1,1

Comments

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.

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.

a(18) has 149 digits.

References

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

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

Links

Daran Gill, Table of n, a(n) for n = 1..19

Example

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.

Mathematica

a={2}; q=1;
For[n=2, n<=12, n++,
    q=q*Last[a];
    AppendTo[a, Min[Select[FactorInteger[q+1][[All, 1]], Mod[#, 4]==3 &]]];
    ];
a (* Robert Price, Jul 16 2015 *)

Crossrefs

Sequence in context: A085682 A267505 A267506 * A241166 A083369 A000946

Adjacent sequences: A218464 A218465 A218466 * A218468 A218469 A218470

Keyword

nonn

Author

Daran Gill, Mar 26 2013

Status

approved