

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
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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](* JeanFrançois 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



