%I #27 Oct 28 2021 20:15:03

%S 2,4,8,16,24,32,64,128,256,320,512,1024,2048,3712,4096,7168,8192,

%T 10512,16192,16384,32768,33024,37888,41728,49280,51552,54528,57280,

%U 62592,65536,66432,67968,68832,69792,81600,84352,87696,91968,92016,93888,94720,124128,129888,131072

%N Integers m that satisfy tau(m) + omega(m) = #({phi(x) = m}).

%C All even terms of A000079 are contained in this sequence.

%C a(5) = 24 is the first term not a term of A000079, a(10) = 320 is the second.

%H Robert Israel, <a href="/A305656/b305656.txt">Table of n, a(n) for n = 1..1000</a>

%H Max Alekseyev, <a href="http://home.gwu.edu/~maxal/gpscripts/">PARI scripts for various problems</a> (see invphi.gp there).

%F tau(m) + omega(m) = #({phi(x) = m}).

%F Integers m such that A163523(m) = A014197(m).

%e 2 is a term because tau(2) = 2, omega(2) = 1, and #({phi(x) = 2}) = 3.

%e 24 is a term because tau(24) = 8, omega(24) = 2, and #({phi(x) = 24}) = 10.

%p filter:= proc(n) uses numtheory; tau(n)+nops(factorset(n)) = nops(invphi(n)) end proc:

%p select(filter, [seq(i,i=2..10^5,2)]); # _Robert Israel_, Oct 28 2021

%t Block[{nn = 10^5, s}, s = Function[s, Function[t, Take[#, nn] &@ ReplacePart[t, Map[# -> Length@ Lookup[s, #] &, Keys@ s]]]@ ConstantArray[0, Max@ Keys@ s]]@ KeySort@ PositionIndex@ Array[EulerPhi, Floor[nn^(3/2)] + 10]; Select[Range@ nn, DivisorSigma[0, #] + PrimeNu[#] == s[[#]] &] ] (* _Michael De Vlieger_, Jul 21 2018 *)

%o (PARI) isok(m) = numdiv(m) + omega(m) == #invphi(m); \\ _Michel Marcus_, Jun 08 2018

%Y Cf. A000005, A001222, A163523.

%Y Cf. A000010, A058277, A014197.

%Y Cf. A000079.

%K nonn

%O 1,1

%A _Torlach Rush_, Jun 07 2018

%E More terms from _Michel Marcus_, Jun 08 2018