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
KEYWORD
nonn,nice
AUTHOR
EXTENSIONS
Description and initial term modified Jan 15 1996
More terms from Jason Earls, Jun 14 2002
STATUS
approved