A002312 Arc-cotangent reducible numbers or non-Størmer numbers: largest prime factor of k^2 + 1 is less than 2*k.

3, 7, 8, 13, 17, 18, 21, 30, 31, 32, 38, 41, 43, 46, 47, 50, 55, 57, 68, 70, 72, 73, 75, 76, 83, 91, 93, 98, 99, 100, 105, 111, 112, 117, 119, 122, 123, 128, 129, 132, 133, 142, 144, 155, 157, 162, 172, 173, 174, 177, 182, 183, 185, 187, 189, 191, 192, 193, 200

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• nice

Comments

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

References

• John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 247.
• 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, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
• J. Todd, Table of Arctangents. National Bureau of Standards, Washington, DC, 1951, p. 94.

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• Graham Everest and Glyn Harman, On primitive divisors of n^2 + b, arXiv:math/0701234 [math.NT], 2007.
• E. Kowalski, On the "reducibility" of arctangents of integers, Amer. Math. Monthly, Vol. 111, No. 4 (Apr. 2004), 351-354.
• Olga Taussky, Sums of Squares, The American Mathematical Monthly, Vol. 77, No. 8 (Oct., 1970), pp. 805-830 (26 pages). See p. 823.
• J. Todd, A problem on arc tangent relations, Amer. Math. Monthly, 56 (1949), 517-528.

Crossrefs

Cf. A005528.

Cf. A006530, A071931 (subsequence).

Programs

Haskell

a002312 n = a002312_list !! (n-1)
a002312_list = filter (\x -> 2 * x > a006530 (x ^ 2 + 1)) [1..]
-- Reinhard Zumkeller, Jun 12 2015

Mathematica

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 *)
Select[Range[200],FactorInteger[#^2+1][[-1,1]]<2#&] (* Harvey P. Dale, Dec 07 2015 *)

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, 201)))) # Michael S. Branicky, Aug 30 2021

Extensions

Description and initial term modified Jan 15 1996

More terms from Jason Earls, Jun 14 2002