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

%I A318145 #24 Mar 13 2020 16:59:42
%S A318145 6,12,14,20,24,28,30,40,48,56,60,62,70,72,80,84,96,112,120,124,126,
%T A318145 132,140,144,168,176,192,198,208,224,240,248,252,254,260,272,286,288,
%U A318145 320,336,340,344,384,390,396,408,430,448,456,480,496,504,508,510,532
%N A318145 Numbers m such that 2^phi(m) mod m is a perfect power other than 1.
%C A318145 All terms are even, as 2^phi(m) == 1 (mod m) if m is odd. - _Robert Israel_, Sep 02 2018
%C A318145 Perfect power terms are 144, 576, 900, 1600, 3136, 9216, 12544, 20736, 36864, 57600, 63504, ... - _Altug Alkan_, Sep 04 2018
%H A318145 Robert Israel, <a href="/A318145/b318145.txt">Table of n, a(n) for n = 1..10000</a>
%p A318145 ispow:= proc(n) local F;
%p A318145   F:= map(t -> t[2], ifactors(n)[2]);
%p A318145   igcd(op(F)) > 1
%p A318145 end proc:
%p A318145 select(m -> ispow(2 &^ numtheory:-phi(m) mod m), [seq(i,i=2..1000,2)]); # _Robert Israel_, Sep 02 2018
%t A318145 okQ[n_] := GCD @@ FactorInteger[PowerMod[2, EulerPhi[n], n]][[All, 2]] > 1;
%t A318145 Select[Range[2, 1000, 2], okQ] (* _Jean-François Alcover_, Aug 02 2019 *)
%o A318145 (Sage)
%o A318145 def isA318145(n):
%o A318145     m = power_mod(2, euler_phi(n), n)
%o A318145     return m > 0 and m.is_perfect_power()
%o A318145 def A318145_list(search_bound):
%o A318145     return [n for n in range(2, search_bound + 1, 2) if isA318145(n)]
%o A318145 print(A318145_list(532))
%Y A318145 Cf. A000010, A001597, A318623. Contains A139257.
%K A318145 nonn
%O A318145 1,1
%A A318145 _Peter Luschny_, Sep 01 2018
%E A318145 Definition corrected by _Robert Israel_, Sep 02 2018