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A260961 Numbers n such that phi(n') = phi(n)', where phi(n) is the Euler totient function of n and n' is the arithmetic derivative of n.

%I A260961 #12 Oct 27 2023 22:14:54
%S A260961 1,3,8,28,32,36,40,50,56,80,128,176,184,228,324,368,448,472,504,560,
%T A260961 576,664,704,1328,1336,1512,1620,1686,1816,2048,2544,2580,2864,3008,
%U A260961 3064,3540,3776,3832,3888,4024,5184,5744,6048,6904,7096,7128,8192,9216,10264,10456
%N A260961 Numbers n such that phi(n') = phi(n)', where phi(n) is the Euler totient function of n and n' is the arithmetic derivative of n.
%H A260961 Alois P. Heinz, <a href="/A260961/b260961.txt">Table of n, a(n) for n = 1..1000</a> (first 200 terms from Paolo P. Lava)
%e A260961 Arithmetic derivative of 56 is 92 and phi(92) = 44, while phi(56) = 24 and the arithmetic derivative of 24 is 44.
%p A260961 with(numtheory): P:=proc(q) local a,b,c,n,p; for n from 1 to q do
%p A260961 a:=n*add(op(2,p)/op(1,p),p=ifactors(n)[2]); b:=phi(n);
%p A260961 c:=b*add(op(2,p)/op(1,p),p=ifactors(b)[2]);
%p A260961 if phi(a)=c then print(n); fi; od; end: P(10^9);
%t A260961 f[n_] := If[Abs@ n < 2, 0, n Total[#2/#1 & @@@ FactorInteger@ Abs@ n]]; Select[Range@ 12000, EulerPhi@ f@ # == f@ EulerPhi@ # &] (* _Michael De Vlieger_, Aug 07 2015, after _Michael Somos_ at A003415 *)
%Y A260961 Cf. A000010, A003415, A190402.
%K A260961 nonn,easy
%O A260961 1,2
%A A260961 _Paolo P. Lava_, Aug 06 2015