A290002 Numbers k such that psi(phi(k)) = phi(psi(k)).

1, 10, 18, 20, 36, 40, 54, 70, 72, 78, 80, 108, 110, 140, 144, 156, 160, 162, 174, 198, 216, 220, 222, 230, 234, 246, 280, 288, 294, 312, 320, 324, 348, 396, 414, 426, 432, 438, 440, 444, 450, 460, 468, 470, 486, 492, 534, 560, 576, 588, 594, 624, 640, 648, 666, 696, 702, 770, 792, 828, 846, 852

Offset

1,2

Comments

Squarefree terms are 1, 10, 70, 78, 110, 174, 222, 230, 246, 426, 438, ...

Common terms of this sequence and A033632 are 1, 14406, 544500, 141118050, ...

From Robert Israel, Sep 03 2017: (Start)

Includes 2^i*3^j if i >= 1 and j >= 2, i.e., 3*A033845, and A020714(n) for n >= 1.

If an even number m is in the sequence, then so is 2*m.

Are there any odd terms other than 1? (End)

a(1) = 1 is the only odd term. LHS of equation allows for 1 and 3 but only for k <= 6. RHS allows for 1 and only for k = 1. - Torlach Rush, Jul 28 2023

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Maple

psi:= proc(n)  n*mul((1+1/i[1]), i=ifactors(n)[2]) end:
select(psi @ numtheory:-phi = numtheory:-phi @ psi, [$1..1000]); # Robert Israel, Sep 03 2017

Mathematica

f[n_] := n Sum[MoebiusMu[d]^2/d, {d, Divisors@ n}]; Select[Range[10^3], f[EulerPhi@ #] == EulerPhi[f@ #] &] (* Michael De Vlieger, Sep 03 2017 *)

Prog

(PARI) a001615(n) = my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1));
isok(n) = eulerphi(a001615(n))==a001615(eulerphi(n)); \\ after Charles R Greathouse IV at A001615

Crossrefs

Cf. A000010, A001615, A020714, A033632, A033845.

Sequence in context: A043750 A043758 A043767 * A072992 A055481 A213717

Adjacent sequences: A289999 A290000 A290001 * A290003 A290004 A290005

Keyword

nonn,easy

Author

Altug Alkan, Sep 03 2017

Status

approved