A120629 Numbers k with property that -k is not a perfect power and the squarefree part of -k is not congruent to 1 modulo 4.

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

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-François 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