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.

5, 101, 1020101, 53, 29, 2507707213238852620996901, 449, 13, 8693, 1997, 6029, 61, 3181837, 113, 181, 1934689, 6143090225314378441493352126119201470973493456817556328833988172277, 4733, 3617, 41, 68141, 37, 51473, 17, 821, 598201519454797, 157, 9689, 2357, 757, 149, 293, 5261

Offset: 1

Author

Labos Elemer, Oct 09 2000

Keywords

• nonn

Comments

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

Examples

			a(4)=53 is the smallest prime divisor of 4*(5.101.1020101)^2+1 = 1061522231810040101 = 53*1613*12417062216309.
		

References

• P. G. L. Dirichlet (1871): Vorlesungen über Zahlentheorie. Braunschweig, Viewig, Supplement VI, 24 pages.
• G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, page 13.

Links

• Daran Gill, Table of n, a(n) for n = 1..33
• Mersenne Forum, Sequence A057207
• OEIS wiki, OEIS sequences needing factors

Crossrefs

Cf. A000945, A000946, A005265, A005266, A051308-A051335, A002144, A057204-A057208.

Programs

Mathematica

t = {5}; Do[q = Times @@ t; AppendTo[t, FactorInteger[1 + 4*q^2][[1, 1]]], {6}]; t (* T. D. Noe, Mar 27 2013 *)

Extensions

Eight more terms, a(9)-a(16), from Max Alekseyev, Apr 27 2009

Seventeen more terms, a(17)-a(33), added by Daran Gill, Mar 23 2013