A002312 Arc-cotangent reducible numbers or non-Størmer numbers: largest prime factor of k^2 + 1 is less than 2*k. (Formerly M2613 N1033)

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

1,1

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

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.

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 *)

Prog

(PARI) is(n)=my(f=factor(n^2+1)[, 1]); f[#f]<2*n \\ Charles R Greathouse IV, Nov 14 2014
(Haskell)
a002312 n = a002312_list !! (n-1)
a002312_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, 201)))) # Michael S. Branicky, Aug 30 2021

Crossrefs

Cf. A005528.

Cf. A006530, A071931 (subsequence).

Sequence in context: A244592 A010342 A108873 * A252496 A279517 A106474

Adjacent sequences: A002309 A002310 A002311 * A002313 A002314 A002315

Keyword

nonn,nice

Author

N. J. A. Sloane

Extensions

Description and initial term modified Jan 15 1996

More terms from Jason Earls, Jun 14 2002

Status

approved