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

Author

Gerard P. Michon, Jun 20 2006

Keywords

• easy
• nice
• nonn

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.

Examples

			-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.
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• G. P. Michon, Artin's Constant.

Crossrefs

Cf. A085397, A005596.

Programs

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 *)