Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%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