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A005528 Størmer numbers or arc-cotangent irreducible numbers: numbers k such that the largest prime factor of k^2 + 1 is >= 2*k.
(Formerly M0950)
10

%I M0950 #52 Aug 30 2021 10:53:53

%S 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 36,37,39,40,42,44,45,48,49,51,52,53,54,56,58,59,60,61,62,63,64,65,66,

%U 67,69,71,74,77,78,79,80,81,82,84,85,86,87,88,89,90,92,94,95,96

%N Størmer numbers or arc-cotangent irreducible numbers: numbers k such that the largest prime factor of k^2 + 1 is >= 2*k.

%C 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 Named after the Norwegian mathematician and astrophysicist Carl Størmer (1874-1957). - _Amiram Eldar_, Jun 08 2021

%D John H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, p. 246.

%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 and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%D John Todd, Table of Arctangents, National Bureau of Standards, Washington, DC, 1951, p. 2.

%H 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 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 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 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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/StormerNumber.html">Størmer Number</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/St%C3%B8rmer_number">Størmer number</a>.

%t Select[Range[96], FactorInteger[#^2 + 1][[-1, 1]] >= 2 # &] (* _Jean-François Alcover_, Apr 11 2011 *)

%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 a005528 n = a005528_list !! (n-1)

%o a005528_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, 97)))) # _Michael S. Branicky_, Aug 30 2021

%Y Cf. A002312, A006530.

%Y Cf. A084925 (hyperbolic analog).

%K nonn,nice,easy

%O 1,2

%A _N. J. A. Sloane_ and _J. H. Conway_

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Last modified March 29 08:13 EDT 2024. Contains 371265 sequences. (Running on oeis4.)