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A002312
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Arc-cotangent reducible numbers or non-Størmer numbers: largest prime factor of k^2 + 1 is less than 2*k.
(Formerly M2613 N1033)
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8
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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
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OFFSET
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1,1
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COMMENTS
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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
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REFERENCES
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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.
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LINKS
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Olga Taussky, Sums of Squares, The American Mathematical Monthly, Vol. 77, No. 8 (Oct., 1970), pp. 805-830 (26 pages). See p. 823.
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MATHEMATICA
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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 *)
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PROG
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(Haskell)
a002312 n = a002312_list !! (n-1)
a002312_list = filter (\x -> 2 * x > a006530 (x ^ 2 + 1)) [1..]
(Python)
from sympy import factorint
def ok(n): return max(factorint(n*n + 1)) < 2*n
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CROSSREFS
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KEYWORD
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nonn,nice
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AUTHOR
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EXTENSIONS
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Description and initial term modified Jan 15 1996
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STATUS
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approved
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