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)

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, 36, 37, 39, 40, 42, 44, 45, 48, 49, 51, 52, 53, 54, 56, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 69, 71, 74, 77, 78, 79, 80, 81, 82, 84, 85, 86, 87, 88, 89, 90, 92, 94, 95, 96

Offset

1,2

Comments

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

Named after the Norwegian mathematician and astrophysicist Carl Størmer (1874-1957). - Amiram Eldar, Jun 08 2021

References

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

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.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

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

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)

Graham Everest and Glyn Harman, On primitive divisors of n^2 + b, arXiv:math/0701234 [math.NT], 2007.

Carl Størmer, 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, Archiv for mathematik og naturvidenskab, Vol. 19, No. 3 (1896), pp. 1-96.

John Todd, A problem on arc tangent relations, Amer. Math. Monthly, 56 (1949), 517-528.

Eric Weisstein's World of Mathematics, Størmer Number.

Wikipedia, Størmer number.

Mathematica

Select[Range[96], FactorInteger[#^2 + 1][[-1, 1]] >= 2 # &] (* Jean-François Alcover, Apr 11 2011 *)

Prog

(PARI) is(n)=my(f=factor(n^2+1)[, 1]); f[#f]>=2*n \\ Charles R Greathouse IV, Nov 14 2014
(Haskell)
a005528 n = a005528_list !! (n-1)
a005528_list = filter (\x -> 2 * x <= a006530 (x ^ 2 + 1)) [1..]
-- Reinhard Zumkeller, Jun 12 2015
(Python)
from sympy import factorint
def ok(n): return max(factorint(n*n + 1)) >= 2*n
print(list(filter(ok, range(1, 97)))) # Michael S. Branicky, Aug 30 2021

Crossrefs

Cf. A002312, A006530.

Cf. A084925 (hyperbolic analog).

Sequence in context: A143070 A340698 A206926 * A211030 A050015 A153218

Adjacent sequences: A005525 A005526 A005527 * A005529 A005530 A005531

Keyword

nonn,nice,easy

Author

N. J. A. Sloane and J. H. Conway

Status

approved