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%I #19 May 17 2017 05:36:15
%S 1,2,3,5,7,8,18,57,239
%N Numbers n such that n^2 + 1 is 13-smooth.
%C Equivalently: Numbers n such that all prime factors of n^2 + 1 are <= 13.
%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*5^a*13^b.
%C This sequence is complete by a theorem of Størmer.
%C The largest instance 239^2 + 1 = 2*13^4 also gives the only nontrivial solution for x^2 + 1 = 2y^4 (Ljunggren).
%D W. Ljunggren, Zur Theorie der Gleichung x^2 + 1 = 2y^4, Avh. Norsk Vid. Akad. Oslo. 1(5) (1942), 1--27.
%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 Ray Steiner, <a href="https://doi.org/10.1016/S0022-314X(05)80029-0">Simplifying the Solution of Ljunggren's Equation X^2 + 1 = 2Y^4</a>, J. Number Theory 37 (1991), 123--132, more accesible proof of Ljunggren's result.
%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 For n = 8, a(8)^2 + 1 = 57^2 + 1 = 3250 = 2*5^3*13.
%t Select[Range[1000], FactorInteger[#^2 + 1][[-1, 1]] <= 13&] (* _Jean-François Alcover_, May 17 2017 *)
%o (PARI) for(n=1, 9e6, if(vecmax(factor(n^2+1)[, 1])<=13, print1(n", ")))
%o (Python)
%o from sympy import primefactors
%o def ok(n): return max(primefactors(n**2 + 1))<=13 # _Indranil Ghosh_, Apr 16 2017
%Y Cf. A014442, A252493 (n(n+1) instead of n^2 + 1).
%K nonn,fini,full
%O 1,2
%A _Tomohiro Yamada_, Apr 16 2017
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