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

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

Author

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

Keywords

• nonn
• nice
• easy

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.

Crossrefs

Cf. A002312, A006530.

Cf. A084925 (hyperbolic analog).

Programs

Haskell

a005528 n = a005528_list !! (n-1)
a005528_list = filter (\x -> 2 * x <= a006530 (x ^ 2 + 1)) [1..]
-- Reinhard Zumkeller, Jun 12 2015

Mathematica

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

PARI

is(n)=my(f=factor(n^2+1)[,1]);f[#f]>=2*n \\ Charles R Greathouse IV, Nov 14 2014

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