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

1,2

Links

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

Example

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

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 *)

Crossrefs

Cf. A000010, A003415, A190402.

Sequence in context: A074271 A005641 A377387 * A230822 A332727 A148845

Adjacent sequences: A260958 A260959 A260960 * A260962 A260963 A260964

Keyword

nonn,easy

Author

Paolo P. Lava, Aug 06 2015

Status

approved