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.

1, 3, 8, 28, 32, 36, 40, 50, 56, 80, 128, 176, 184, 228, 324, 368, 448, 472, 504, 560, 576, 664, 704, 1328, 1336, 1512, 1620, 1686, 1816, 2048, 2544, 2580, 2864, 3008, 3064, 3540, 3776, 3832, 3888, 4024, 5184, 5744, 6048, 6904, 7096, 7128, 8192, 9216, 10264, 10456

Offset: 1

Author

Paolo P. Lava, Aug 06 2015

Keywords

• nonn
• easy

Examples

			Arithmetic derivative of 56 is 92 and phi(92) = 44, while phi(56) = 24 and the arithmetic derivative of 24 is 44.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1000 (first 200 terms from Paolo P. Lava)

Crossrefs

Cf. A000010, A003415, A190402.

Programs

Maple

with(numtheory): P:=proc(q) local a,b,c,n,p; for n from 1 to q do
a:=n*add(op(2,p)/op(1,p),p=ifactors(n)[2]); b:=phi(n);
c:=b*add(op(2,p)/op(1,p),p=ifactors(b)[2]);
if phi(a)=c then print(n); fi; od; end: P(10^9);

Mathematica

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 *)