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A290002
Numbers k such that psi(phi(k)) = phi(psi(k)).
6
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
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
KEYWORD
nonn,easy
AUTHOR
Altug Alkan, Sep 03 2017
STATUS
approved