cp's OEIS Frontend

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

%I A290002 #31 Jul 30 2023 09:23:25
%S A290002 1,10,18,20,36,40,54,70,72,78,80,108,110,140,144,156,160,162,174,198,
%T A290002 216,220,222,230,234,246,280,288,294,312,320,324,348,396,414,426,432,
%U A290002 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
%N A290002 Numbers k such that psi(phi(k)) = phi(psi(k)).
%C A290002 Squarefree terms are 1, 10, 70, 78, 110, 174, 222, 230, 246, 426, 438, ...
%C A290002 Common terms of this sequence and A033632 are 1, 14406, 544500, 141118050, ...
%C A290002 From _Robert Israel_, Sep 03 2017: (Start)
%C A290002 Includes 2^i*3^j if i >= 1 and j >= 2, i.e., 3*A033845, and A020714(n) for n >= 1.
%C A290002 If an even number m is in the sequence, then so is 2*m.
%C A290002 Are there any odd terms other than 1? (End)
%C A290002 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
%H A290002 Robert Israel, <a href="/A290002/b290002.txt">Table of n, a(n) for n = 1..10000</a>
%p A290002 psi:= proc(n)  n*mul((1+1/i[1]), i=ifactors(n)[2]) end:
%p A290002 select(psi @ numtheory:-phi = numtheory:-phi @ psi, [$1..1000]); # _Robert Israel_, Sep 03 2017
%t A290002 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 *)
%o A290002 (PARI) a001615(n) = my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1));
%o A290002 isok(n) = eulerphi(a001615(n))==a001615(eulerphi(n)); \\ after _Charles R Greathouse IV_ at A001615
%Y A290002 Cf. A000010, A001615, A020714, A033632, A033845.
%K A290002 nonn,easy
%O A290002 1,2
%A A290002 _Altug Alkan_, Sep 03 2017