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

Author

Altug Alkan, Sep 03 2017

Keywords

• nonn
• easy

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

Crossrefs

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

Programs

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 *)

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