

A292544


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


5



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
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OFFSET

1,2


COMMENTS

Conjecture: For n > 1, a(n) is a Zumkeller number (A083207) [confirmed for n up to 47].  Ivan N. Ianakiev, Sep 22 2017


LINKS



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*i1) 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*i2) is a term for all i >= 1.
If m = 77, then z = 30 and k = 14 ==> 77*2^(30*i13) is a term for all i >= 1.


EXAMPLE

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.


MATHEMATICA

{1}~Join~Select[Range[10^6], Function[n, # == PowerMod[2, #, n] &@ EulerPhi@ n]] (* Michael De Vlieger, Sep 18 2017 *)


PROG

(PARI) isok(n) = Mod(2, n)^eulerphi(n)==eulerphi(n);


CROSSREFS



KEYWORD

nonn,easy


AUTHOR



STATUS

approved



