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%I #7 Apr 20 2017 14:53:58
%S 1,4,9,15,22,32,41,57,74,94,120,156
%N Number of integers x such that the greatest prime factor of x^2 + 1 is at most A002313(n), the n-th prime not congruent to 3 mod 4.
%C In other words, x^2 + 1 is A002313(n)-smooth.
%C Størmer shows that the number of such integers is finite for any n.
%C a(n) <= 3^n - 2^n follows from Størmer's argument.
%C a(n) <= (2^n-1)*(A002313(n)+1)/2 is implicit in Lehmer 1964.
%C Luca 2004 determines all integers x such that x^2 + 1 is 100-smooth, which is pushed to 200 by Najman 2010.
%H D. H. Lehmer, <a href="http://projecteuclid.org/euclid.ijm/1256067456">On a problem of Størmer</a>, Ill. J. Math., 8 (1964), 57--69.
%H Florian Luca, <a href="http://www.emis.de/journals/AMI/2004/acta2004-luca.pdf">Primitive divisors of Lucas sequences and prime factors of x^2 + 1 and x^4 + 1</a>, Acta Academiae Paedagogicae Agriensis, Sectio Mathematicae 31 (2004), pp. 19--24.
%H Filip Najman, <a href="http://doi.org/10.3336/gm.45.2.04">Smooth values of some quadratic polynomials</a>, Glas. Mat. 45 (2010), 347--355. Tables are available in the author's <a href="http://web.math.hr/~fnajman/">Home Page</a> (gives all 811 numbers x such that x^2+1 has no prime factor greater than 197).
%H A. Schinzel, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa13/aa13113.pdf">On two theorems of Gelfond and some of their applications</a>, Acta Arithmetica 13 (1967-1968), 177--236.
%H Carl Størmer, <a href="http://www.archive.org/stream/skrifterudgivnea1897chri#page/n79/mode/2up">Quelques théorèmes sur l'équation de Pell x^2 - Dy^2 = +-1 et leurs applications</a> (in French), Skrifter Videnskabs-selskabet (Christiania), Mat.-Naturv. Kl. I (2), 48 pp.
%Y Equivalents for x(x+1): A145604
%Y Cf. A014442, A185389
%Y Cf. A285282.
%K nonn,hard,more
%O 1,2
%A _Tomohiro Yamada_, Apr 16 2017
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