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

%I A078148 #32 Mar 18 2025 07:28:42
%S A078148 1,2,4,6,16,24,30,36,64,384,408,480,510,1024,1296,1560,1680,2304,2640,
%T A078148 3480,4096,5440,5520,6360,9240,11280,14040,14160,14400,15120,15960,
%U A078148 17880,19320,19920,20760,22848,24480,25680,26880,30360,32280,35160
%N A078148 Numbers k such that d(phi(k)) = phi(d(k)), where d=A000005 and phi=A000010.
%C A078148 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
%C A078148 Next odd term after 1 is 15230439315 (cf. A378315). - _Max Alekseyev_, Jan 09 2025
%H A078148 Amiram Eldar, <a href="/A078148/b078148.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..500 from T. D. Noe)
%e A078148 k = 24: d(24) = 8, phi(8) = 4, phi(24) = 8, d(8) = 4, so 24 is a term.
%t A078148 cm[x_] := DivisorSigma[0, EulerPhi[x]]-EulerPhi[DivisorSigma[0, x]] Do[s=cm[n]; If[Equal[s, 0], Print[n]], {n, 1, 100000}]
%t A078148 Select[Range[36000],DivisorSigma[0,EulerPhi[#]]==EulerPhi[ DivisorSigma[ 0,#]]&] (* _Harvey P. Dale_, Sep 02 2013 *)
%o A078148 (PARI) is(n)=numdiv(eulerphi(n))==eulerphi(numdiv(n)) \\ _Charles R Greathouse IV_, Feb 21 2013
%Y A078148 The odd terms are listed in A378315.
%Y A078148 Cf. A000005, A000010, A033632.
%K A078148 nonn
%O A078148 1,2
%A A078148 _Labos Elemer_, Nov 26 2002