Search: id:A083844
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%I A083844
%S A083844 2,4,10,19,51,112,316,841,2378,6656,18822,54110,156081,456362,1339875,
%T A083844 3954181,11726896,34900213,104248948,312357934,938457801,2826683630
%N A083844 Number of primes of the form x^2 + 1 < 10^n.
%C A083844 It is conjectured that this sequence is infinite, but this has never 
               been proved.
%C A083844 These primes can be found quickly using a sieve based on the fact that 
               numbers of this form have at most one primitive prime factor (A005529). 
               The sum of the reciprocals of these primes is 0.81459657... - T. 
               D. Noe (noe(AT)sspectra.com), Oct 14 2003
%D A083844 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 
               5th ed., Oxford Univ. Press, 1979, th. 17.
%D A083844 P. Ribenboim, The Little Book of Big Primes. Springer-Verlag, 1991, p. 
               190.
%H A083844 C. K. Caldwell, <a href="http://www.utm.edu/~caldwell/preprints/Heuristics.pdf">
               AN AMAZING PRIME HEURISTIC</a> A pdf file.
%H A083844 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               LandausProblems.html">Link to a section of The World of Mathematics.</
               a> Landau's Problems.
%H A083844 Marek Wolf, <a href="http://arXiv.org/abs/0803.1456">Search for primes 
               of the form m^2+1</a>
%e A083844 a(3) = 10 because the only primes or the form x^2 + 1 < 10^3 are the
%e A083844 ten primes: 2, 5, 17, 37, 101, 197, 257, 401, 577 & 677.
%t A083844 c = 1; k = 2; (* except for the initial prime 2, all X's must be odd. 
               *) Do[ While[ k^2 + 1 < 10^n, If[ PrimeQ[k^2 + 1], c++ ]; k += 2]; 
               Print[c], {n, 1, 20}]
%Y A083844 Cf. A005574, A002496, A083845, A083846, A083847, A083848, A083849.
%Y A083844 Cf. A005529 (primitive prime factors of the sequence k^2+1).
%Y A083844 Sequence in context: A079162 A043330 A011963 this_sequence A026554 A099413 
               A127392
%Y A083844 Adjacent sequences: A083841 A083842 A083843 this_sequence A083845 A083846 
               A083847
%K A083844 nonn
%O A083844 1,1
%A A083844 Harry J. Smith (hjsmithh(AT)sbcglobal.net), May 05 2003
%E A083844 Edited by Robert G. Wilson v (rgwv(AT)rgwv.com), May 08 2003
%E A083844 More terms from T. D. Noe (noe(AT)sspectra.com), Oct 14 2003
%E A083844 a(17)-a(22) from Robert Gerbicz (robert.gerbicz(AT)gmail.com), Apr 15 
               2009

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