A078148 Numbers k such that d(phi(k)) = phi(d(k)), where d=A000005 and phi=A000010.

1, 2, 4, 6, 16, 24, 30, 36, 64, 384, 408, 480, 510, 1024, 1296, 1560, 1680, 2304, 2640, 3480, 4096, 5440, 5520, 6360, 9240, 11280, 14040, 14160, 14400, 15120, 15960, 17880, 19320, 19920, 20760, 22848, 24480, 25680, 26880, 30360, 32280, 35160

Offset: 1

Author

Labos Elemer, Nov 26 2002

Keywords

• nonn

Comments

2^m is in the sequence iff m=0 or m+1 is prime (the proof is easy). Also all numbers of the form 3*2^(2^m-1) are in the sequence because d(phi(3*2^(2^m-1))) - phi(d(3*2^(2^m-1)))= d(2*2^(2^m-2)) - phi(2*2^m) = d(2^(2^m-1)) - phi(2^(m+1)) = 2^m - 2^m = 0. So this sequence is infinite. - Farideh Firoozbakht, Jan 25 2006

Next odd term after 1 is 15230439315 (cf. A378315). - Max Alekseyev, Jan 09 2025

Examples

			k = 24: d(24) = 8, phi(8) = 4, phi(24) = 8, d(8) = 4, so 24 is a term.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..500 from T. D. Noe)

Crossrefs

The odd terms are listed in A378315.

Cf. A000005, A000010, A033632.

Programs

Mathematica

cm[x_] := DivisorSigma[0, EulerPhi[x]]-EulerPhi[DivisorSigma[0, x]] Do[s=cm[n]; If[Equal[s, 0], Print[n]], {n, 1, 100000}]
Select[Range[36000],DivisorSigma[0,EulerPhi[#]]==EulerPhi[ DivisorSigma[ 0,#]]&] (* Harvey P. Dale, Sep 02 2013 *)

PARI

is(n)=numdiv(eulerphi(n))==eulerphi(numdiv(n)) \\ Charles R Greathouse IV, Feb 21 2013