A292544 Numbers h such that 2^phi(h) == phi(h) (mod h).
1, 12, 40, 48, 60, 192, 544, 640, 680, 704, 768, 816, 960, 1020, 1664, 3072, 10240, 11008, 12288, 13760, 15360, 19456, 24320, 49152, 83968, 125952, 131584, 139264, 139808, 163840, 164480, 174080, 174760, 196608, 197376, 208896, 209712, 245760, 246720, 261120, 262140, 720896, 786432
Offset: 1
Examples
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.
Links
- Giovanni Resta, Table of n, a(n) for n = 1..180 (terms < 10^12; first 101 terms from Michel Marcus)
Programs
-
Mathematica
{1}~Join~Select[Range[10^6], Function[n, # == PowerMod[2, #, n] &@ EulerPhi@ n]] (* Michael De Vlieger, Sep 18 2017 *) -
PARI
isok(n) = Mod(2, n)^eulerphi(n)==eulerphi(n);
Formula
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.
Some families of solutions of the form m*2^(i*z - k + 1):
If m = 3, then z = 2 and k = 1 ==> 3*2^(2*i) is a term for all i >= 1.
If m = 5, then z = 4 and k = 2 ==> 5*2^(4*i-1) is a term for all i >= 1.
If m = 7, then z = 3 but k does not exist ==> no term with odd part equal to 7.
If m = 15, then z = 4 and k = 3 ==> 15*2^(4*i-2) is a term for all i >= 1.
If m = 77, then z = 30 and k = 14 ==> 77*2^(30*i-13) is a term for all i >= 1.
Comments