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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. 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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

REFERENCES

P. G. L. Dirichlet (1871): Vorlesungen ├╝ber 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

EXAMPLE

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

MATHEMATICA

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

CROSSREFS

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

Sequence in context: A009765 A113073 A275749 * A124986 A231830 A260024

Adjacent sequences:  A057204 A057205 A057206 * A057208 A057209 A057210

KEYWORD

nonn

AUTHOR

Labos Elemer, Oct 09 2000

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

STATUS

approved

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Last modified June 4 03:40 EDT 2020. Contains 334815 sequences. (Running on oeis4.)