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A217715 Numbers equal to the Euler totient function of their arithmetic derivative: k = phi(k').

%I A217715 #32 Apr 30 2022 08:22:07
%S A217715 16,30,32,48,54,64,80,120,176,288,368,432,464,656,848,858,864,1328,
%T A217715 1424,1722,1808,1944,2096,2768,2864,2916,3056,3728,3824,4016,4496,
%U A217715 4688,5744,5832,6704,6896,7088,7856,8144,9488,10256,10448,10544,10928,11504,11888
%N A217715 Numbers equal to the Euler totient function of their arithmetic derivative: k = phi(k').
%C A217715 If p is a Sophie Germain prime (A005384) then m = 16*p is a term. Indeed: m' = (16*p)' = 32*p + 16 = 16*(2*p + 1) and phi(m') = phi(32*p + 16) = phi(16*(2*p + 1)) = 8*phi(2*p + 1) = 8*2*p = m for odd p. If p = 2 then m = 16*2 = 32 is a term. - _Marius A. Burtea_, Apr 10 2022
%H A217715 Amiram Eldar, <a href="/A217715/b217715.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..200 from Paolo P. Lava)
%e A217715 For k=368, k'=752 and phi(752)=368.
%p A217715 with(numtheory);
%p A217715 A217715:= proc(q) local n,p;
%p A217715 for n from 1 to q do
%p A217715 if phi(n*add(op(2,p)/op(1,p),p=ifactors(n)[2]))=n then print(n); fi; od; end:
%p A217715 A217715(10^6);
%t A217715 aQ[1]=1; aQ[n_] := EulerPhi[n * Total[#2/#1 & @@@ FactorInteger[n]]] == n; Select[Range[10000], aQ] (* _Amiram Eldar_, Jul 11 2019 *)
%o A217715 (Magma) f:=func<n | n le 1 select 0 else  n*(&+[Factorisation(n)[i][2] / Factorisation(n)[i][1]: i in [1..#Factorisation(n)]])>;  [k:k in [2..12000]| k eq EulerPhi(Floor(f(k)))]; // _Marius A. Burtea_, Apr 09 2022
%Y A217715 Cf. A000010, A003415, A005384, A166374.
%K A217715 nonn
%O A217715 1,1
%A A217715 _Paolo P. Lava_, Mar 21 2013