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%I A005528 M0950 #52 Aug 30 2021 10:53:53
%S A005528 1,2,4,5,6,9,10,11,12,14,15,16,19,20,22,23,24,25,26,27,28,29,33,34,35,
%T A005528 36,37,39,40,42,44,45,48,49,51,52,53,54,56,58,59,60,61,62,63,64,65,66,
%U A005528 67,69,71,74,77,78,79,80,81,82,84,85,86,87,88,89,90,92,94,95,96
%N A005528 Størmer numbers or arc-cotangent irreducible numbers: numbers k such that the largest prime factor of k^2 + 1 is >= 2*k.
%C A005528 Also numbers k such that k^2 + 1 has a primitive divisor, hence (by Everest & Harman, Theorem 1.4) 1.1n < a(n) < 1.88n for large enough n. They conjecture that a(n) ~ cn where c = 1/log 2 = 1.4426.... - _Charles R Greathouse IV_, Nov 15 2014
%C A005528 Named after the Norwegian mathematician and astrophysicist Carl Størmer (1874-1957). - _Amiram Eldar_, Jun 08 2021
%D A005528 John H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, p. 246.
%D A005528 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 A005528 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A005528 John Todd, Table of Arctangents, National Bureau of Standards, Washington, DC, 1951, p. 2.
%H A005528 Amiram Eldar, <a href="/A005528/b005528.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from T. D. Noe)
%H A005528 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 A005528 Carl Størmer, <a href="https://archive.org/details/archivformathema1918961897oslo/page/n121/mode/2up">Sur l'application de la théorie des nombres entiers complexes à la solution en nombres rationnels x_1 x_2... x_n c_1 c_2... c_n, k de l'équation: c_1 arc tg x_1 + c_2 arc tg x_2 + ... + c_n arc tg x_n = k * Pi/4</a>, Archiv for mathematik og naturvidenskab, Vol. 19, No. 3 (1896), pp. 1-96.
%H A005528 John Todd, <a href="http://www.jstor.org/stable/2305526">A problem on arc tangent relations</a>, Amer. Math. Monthly, 56 (1949), 517-528.
%H A005528 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/StormerNumber.html">Størmer Number</a>.
%H A005528 Wikipedia, <a href="https://en.wikipedia.org/wiki/St%C3%B8rmer_number">Størmer number</a>.
%t A005528 Select[Range[96], FactorInteger[#^2 + 1][[-1, 1]] >= 2 # &] (* _Jean-François Alcover_, Apr 11 2011 *)
%o A005528 (PARI) is(n)=my(f=factor(n^2+1)[,1]);f[#f]>=2*n \\ _Charles R Greathouse IV_, Nov 14 2014
%o A005528 (Haskell)
%o A005528 a005528 n = a005528_list !! (n-1)
%o A005528 a005528_list = filter (\x -> 2 * x <= a006530 (x ^ 2 + 1)) [1..]
%o A005528 -- _Reinhard Zumkeller_, Jun 12 2015
%o A005528 (Python)
%o A005528 from sympy import factorint
%o A005528 def ok(n): return max(factorint(n*n + 1)) >= 2*n
%o A005528 print(list(filter(ok, range(1, 97)))) # _Michael S. Branicky_, Aug 30 2021
%Y A005528 Cf. A002312, A006530.
%Y A005528 Cf. A084925 (hyperbolic analog).
%K A005528 nonn,nice,easy
%O A005528 1,2
%A A005528 _N. J. A. Sloane_ and _J. H. Conway_

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