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

16, 30, 32, 48, 54, 64, 80, 120, 176, 288, 368, 432, 464, 656, 848, 858, 864, 1328, 1424, 1722, 1808, 1944, 2096, 2768, 2864, 2916, 3056, 3728, 3824, 4016, 4496, 4688, 5744, 5832, 6704, 6896, 7088, 7856, 8144, 9488, 10256, 10448, 10544, 10928, 11504, 11888

Offset

1,1

Comments

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

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..200 from Paolo P. Lava)

Example

For k=368, k'=752 and phi(752)=368.

Maple

with(numtheory);
A217715:= proc(q) local n, p;
for n from 1 to q do
if phi(n*add(op(2, p)/op(1, p), p=ifactors(n)[2]))=n then print(n); fi; od; end:
A217715(10^6);

Mathematica

aQ[1]=1; aQ[n_] := EulerPhi[n * Total[#2/#1 & @@@ FactorInteger[n]]] == n; Select[Range[10000], aQ] (* Amiram Eldar, Jul 11 2019 *)

Prog

(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

Crossrefs

Cf. A000010, A003415, A005384, A166374.

Sequence in context: A115278 A228664 A359935 * A064634 A255265 A183372

Adjacent sequences: A217712 A217713 A217714 * A217716 A217717 A217718

Keyword

nonn

Author

Paolo P. Lava, Mar 21 2013

Status

approved