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%I #14 Jul 16 2021 03:26:31
%S 2,4,5,6,9,10,13,14,16,17,18,20,21,22,24,25,26,29,30,33,34,36,37,38,
%T 40,41,42,45,46,49,50,52,53,54,56,57,58,61,62,65,66,68,69,70,72,73,74,
%U 77,78,80,81,82,84,85,86,88,89,90,93,94,96,97,98,100,101,102,104,105,106
%N Numbers k with property that -k is not a perfect power and the squarefree part of -k is not congruent to 1 modulo 4.
%C According to a famous 1927 conjecture of Emil Artin, modified by Dick Lehmer, these negative numbers are primitive roots modulo each prime of a set whose density among primes equals Artin's constant (see A005596). The positive numbers with the same property are given by A085397.
%H T. D. Noe, <a href="/A120629/b120629.txt">Table of n, a(n) for n = 1..1000</a>
%H G. P. Michon, <a href="http://www.numericana.com/answer/constants.htm#artin">Artin's Constant</a>.
%e -3 and -12 are not in the set because their squarefree parts are equal to -3, which is congruent to 1 modulo 4. -32 is not in the set because it is the fifth power of -2. -1 is excluded because it is an odd power of -1.
%t SquareFreePart[n_] := Times @@ Apply[ Power, ({#[[1]], Mod[#[[2]], 2]} & ) /@ FactorInteger[n], {1}]; perfectPowerQ[n_] := (r = False; For[k = 2, k <= Abs[n] + 2, k++, If[Reduce[n == x^k, {x}, Integers] =!= False, r = True; Break[]]]; r); ok[n_] := ! perfectPowerQ[-n] && Mod[SquareFreePart[-n], 4] != 1; Select[Range[106], ok](* _Jean-François Alcover_, Feb 14 2012 *)
%Y Cf. A085397, A005596.
%K easy,nice,nonn
%O 1,1
%A _Gerard P. Michon_, Jun 20 2006
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