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A120629 Numbers n with property that -n is not a perfect power and the squarefree part of -n is not congruent to 1 modulo 4. 1
2, 4, 5, 6, 9, 10, 13, 14, 16, 17, 18, 20, 21, 22, 24, 25, 26, 29, 30, 33, 34, 36, 37, 38, 40, 41, 42, 45, 46, 49, 50, 52, 53, 54, 56, 57, 58, 61, 62, 65, 66, 68, 69, 70, 72, 73, 74, 77, 78, 80, 81, 82, 84, 85, 86, 88, 89, 90, 93, 94, 96, 97, 98, 100, 101, 102, 104, 105, 106 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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.

LINKS

T. D. Noe, Table of n, a(n) for n=1..1000

G. P. Michon, Artin's Constant.

EXAMPLE

-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.

MATHEMATICA

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-Fran├žois Alcover, Feb 14 2012 *)

CROSSREFS

Cf. A085397, A005596.

Sequence in context: A143072 A089648 A062861 * A169694 A285163 A015834

Adjacent sequences:  A120626 A120627 A120628 * A120630 A120631 A120632

KEYWORD

easy,nice,nonn

AUTHOR

Gerard P. Michon, Jun 20 2006

STATUS

approved

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Last modified May 26 08:55 EDT 2017. Contains 287093 sequences.