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%I M2613 N1033 #49 Dec 20 2021 20:28:39
%S 3,7,8,13,17,18,21,30,31,32,38,41,43,46,47,50,55,57,68,70,72,73,75,76,
%T 83,91,93,98,99,100,105,111,112,117,119,122,123,128,129,132,133,142,
%U 144,155,157,162,172,173,174,177,182,183,185,187,189,191,192,193,200
%N Arc-cotangent reducible numbers or non-Størmer numbers: largest prime factor of k^2 + 1 is less than 2*k.
%C Also numbers such that k^2 + 1 has no primitive divisor, hence (by Everest & Harman, Theorem 1.4) 2.138n < a(n) < 10.6n for large enough n. They conjecture that a(n) ~ cn where c = 1/(1 - log 2) = 3.258.... - _Charles R Greathouse IV_, Nov 15 2014
%D Graham Everest and Glyn Harman, On primitive divisors of n^2 + b, in Number Theory and Polynomials (James McKee and Chris Smyth, ed.), London Mathematical Society 2008.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D J. Todd, Table of Arctangents. National Bureau of Standards, Washington, DC, 1951, p. 94.
%H T. D. Noe, <a href="/A002312/b002312.txt">Table of n, a(n) for n = 1..1000</a>
%H Graham Everest and Glyn Harman, <a href="http://arxiv.org/abs/math/0701234">On primitive divisors of n^2 + b</a>, arXiv:math/0701234 [math.NT], 2007.
%H E. Kowalski, <a href="http://www.jstor.org/stable/4145245">On the "reducibility" of arctangents of integers</a>, Amer. Math. Monthly, Vol. 111, No. 4 (Apr. 2004), 351-354.
%H Olga Taussky, <a href="https://doi.org/10.2307/2317016">Sums of Squares</a>, The American Mathematical Monthly, Vol. 77, No. 8 (Oct., 1970), pp. 805-830 (26 pages). See p. 823.
%H J. Todd, <a href="http://www.jstor.org/stable/2305526">A problem on arc tangent relations</a>, Amer. Math. Monthly, 56 (1949), 517-528.
%t lst={}; Do[n=m^2+1; p=FactorInteger[n][[ -1, 1]]; If[p<2m, AppendTo[lst, m]], {m, 200}]; lst (* _T. D. Noe_, Apr 09 2004 *)
%t Select[Range[200],FactorInteger[#^2+1][[-1,1]]<2#&] (* _Harvey P. Dale_, Dec 07 2015 *)
%o (PARI) is(n)=my(f=factor(n^2+1)[,1]);f[#f]<2*n \\ _Charles R Greathouse IV_, Nov 14 2014
%o (Haskell)
%o a002312 n = a002312_list !! (n-1)
%o a002312_list = filter (\x -> 2 * x > a006530 (x ^ 2 + 1)) [1..]
%o -- _Reinhard Zumkeller_, Jun 12 2015
%o (Python)
%o from sympy import factorint
%o def ok(n): return max(factorint(n*n + 1)) < 2*n
%o print(list(filter(ok, range(1, 201)))) # _Michael S. Branicky_, Aug 30 2021
%Y Cf. A005528.
%Y Cf. A006530, A071931 (subsequence).
%K nonn,nice
%O 1,1
%A _N. J. A. Sloane_
%E Description and initial term modified Jan 15 1996
%E More terms from _Jason Earls_, Jun 14 2002
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