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

1,2

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

Giovanni Resta, Table of n, a(n) for n = 1..180 (terms < 10^12; first 101 terms from Michel Marcus)

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.

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

Cf. A000010, A007733, A066781.

Sequence in context: A226348 A359023 A139691 * A345924 A114815 A363121

Adjacent sequences: A292541 A292542 A292543 * A292545 A292546 A292547

Keyword

nonn,easy

Author

Max Alekseyev and Altug Alkan, Sep 18 2017

Status

approved