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A057207
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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.
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4
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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
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OFFSET
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1,1
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COMMENTS
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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
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REFERENCES
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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.
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LINKS
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EXAMPLE
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a(4)=53 is the smallest prime divisor of 4*(5.101.1020101)^2+1 = 1061522231810040101 = 53*1613*12417062216309.
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MATHEMATICA
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t = {5}; Do[q = Times @@ t; AppendTo[t, FactorInteger[1 + 4*q^2][[1, 1]]], {6}]; t (* T. D. Noe, Mar 27 2013 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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Seventeen more terms, a(17)-a(33), added by Daran Gill, Mar 23 2013
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STATUS
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approved
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