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A292544 Numbers h such that 2^phi(h) == phi(h) (mod h).

%I A292544 #35 Nov 06 2023 03:21:25
%S A292544 1,12,40,48,60,192,544,640,680,704,768,816,960,1020,1664,3072,10240,
%T A292544 11008,12288,13760,15360,19456,24320,49152,83968,125952,131584,139264,
%U A292544 139808,163840,164480,174080,174760,196608,197376,208896,209712,245760,246720,261120,262140,720896,786432
%N A292544 Numbers h such that 2^phi(h) == phi(h) (mod h).
%C A292544 Conjecture: For n > 1, a(n) is a Zumkeller number (A083207) [confirmed for n up to 47]. - _Ivan N. Ianakiev_, Sep 22 2017
%H A292544 Giovanni Resta, <a href="/A292544/b292544.txt">Table of n, a(n) for n = 1..180</a> (terms < 10^12; first 101 terms from Michel Marcus)
%F A292544 Let m be an odd number, z = A007733(m) and k, 0 <= k < z, be such that phi(m) == 2^k (mod m); then m*2^(i*z - k + 1) belongs to this sequence for all i >= 1. And this is a general form of the terms of this sequence.
%F A292544 Some families of solutions of the form m*2^(i*z - k + 1):
%F A292544 If m = 3, then z = 2 and k = 1 ==> 3*2^(2*i) is a term for all i >= 1.
%F A292544 If m = 5, then z = 4 and k = 2 ==> 5*2^(4*i-1) is a term for all i >= 1.
%F A292544 If m = 7, then z = 3 but k does not exist ==> no term with odd part equal to 7.
%F A292544 If m = 15, then z = 4 and k = 3 ==> 15*2^(4*i-2) is a term for all i >= 1.
%F A292544 If m = 77, then z = 30 and k = 14 ==> 77*2^(30*i-13) is a term for all i >= 1.
%e A292544 704 = 11*2^6 is a term since phi(11*2^6) = 5*2^6 and 11*2^6 divides 2^(5*2^6) - 5*2^6.
%t A292544 {1}~Join~Select[Range[10^6], Function[n, # == PowerMod[2, #, n] &@ EulerPhi@ n]] (* _Michael De Vlieger_, Sep 18 2017 *)
%o A292544 (PARI) isok(n) = Mod(2, n)^eulerphi(n)==eulerphi(n);
%Y A292544 Cf. A000010, A007733, A066781.
%K A292544 nonn,easy
%O A292544 1,2
%A A292544 _Max Alekseyev_ and _Altug Alkan_, Sep 18 2017