A318145 Numbers m such that 2^phi(m) mod m is a perfect power other than 1.

6, 12, 14, 20, 24, 28, 30, 40, 48, 56, 60, 62, 70, 72, 80, 84, 96, 112, 120, 124, 126, 132, 140, 144, 168, 176, 192, 198, 208, 224, 240, 248, 252, 254, 260, 272, 286, 288, 320, 336, 340, 344, 384, 390, 396, 408, 430, 448, 456, 480, 496, 504, 508, 510, 532

Offset: 1

Author

Peter Luschny, Sep 01 2018

Keywords

• nonn

Comments

All terms are even, as 2^phi(m) == 1 (mod m) if m is odd. - Robert Israel, Sep 02 2018

Perfect power terms are 144, 576, 900, 1600, 3136, 9216, 12544, 20736, 36864, 57600, 63504, ... - Altug Alkan, Sep 04 2018

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000010, A001597, A318623. Contains A139257.

Programs

Maple

ispow:= proc(n) local F;
  F:= map(t -> t[2], ifactors(n)[2]);
  igcd(op(F)) > 1
end proc:
select(m -> ispow(2 &^ numtheory:-phi(m) mod m), [seq(i,i=2..1000,2)]); # Robert Israel, Sep 02 2018

Mathematica

okQ[n_] := GCD @@ FactorInteger[PowerMod[2, EulerPhi[n], n]][[All, 2]] > 1;
Select[Range[2, 1000, 2], okQ] (* Jean-François Alcover, Aug 02 2019 *)

Sage

def isA318145(n):
    m = power_mod(2, euler_phi(n), n)
    return m > 0 and m.is_perfect_power()
def A318145_list(search_bound):
    return [n for n in range(2, search_bound + 1, 2) if isA318145(n)]
print(A318145_list(532))

Extensions

Definition corrected by Robert Israel, Sep 02 2018