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%I #32 Apr 23 2017 01:04:35
%S 1,2,3,4,5,6,7,8,9,11,12,13,17,18,21,22,23,27,30,31,32,34,38,41,43,46,
%T 47,50,55,57,68,70,72,73,75,83,99,117,119,123,132,133,157,172,173,182,
%U 191,216,233,239,242,255,265,268,278,302,307,319,327,378,401,411,438,447
%N Numbers n such that n^2 + 1 is 100-smooth
%C Equivalently: Numbers n such that all prime factors of n^2 + 1 are <= 97.
%C Since an odd prime factor of n^2 + 1 must be of the form 4m + 1, n^2 + 1 must be of the form 2^t*5^a*13^b*17^c*29^d*37^e*41^f*53^g*61^h*73^i*89^j*97^k, with t = 0 or 1.
%C Luca determined all terms.
%H Tomohiro Yamada, <a href="/A285523/b285523.txt">Table of n, a(n) for n = 1..156</a>
%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 n such that n^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 Nr. 2 (1897), 48 pp.
%e 157^2 + 1 = 2*5^2*17*29 so 157 is a term.
%o (PARI) isok(n) = vecmax(factor(n^2+1)[,1]) <= 100; \\ _Michel Marcus_, Apr 23 2017
%Y Cf. A285282 (n^2 + 1 is 13-smooth), A285283.
%K nonn,fini,full
%O 1,2
%A _Tomohiro Yamada_, Apr 22 2017
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