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A305656 Integers m that satisfy tau(m) + omega(m) = #({phi(x) = m}).

%I A305656 #27 Oct 28 2021 20:15:03
%S A305656 2,4,8,16,24,32,64,128,256,320,512,1024,2048,3712,4096,7168,8192,
%T A305656 10512,16192,16384,32768,33024,37888,41728,49280,51552,54528,57280,
%U A305656 62592,65536,66432,67968,68832,69792,81600,84352,87696,91968,92016,93888,94720,124128,129888,131072
%N A305656 Integers m that satisfy tau(m) + omega(m) = #({phi(x) = m}).
%C A305656 All even terms of A000079 are contained in this sequence.
%C A305656 a(5) = 24 is the first term not a term of A000079, a(10) = 320 is the second.
%H A305656 Robert Israel, <a href="/A305656/b305656.txt">Table of n, a(n) for n = 1..1000</a>
%H A305656 Max Alekseyev, <a href="http://home.gwu.edu/~maxal/gpscripts/">PARI scripts for various problems</a> (see invphi.gp there).
%F A305656 tau(m) + omega(m) = #({phi(x) = m}).
%F A305656 Integers m such that A163523(m) = A014197(m).
%e A305656 2 is a term because tau(2) = 2, omega(2) = 1, and #({phi(x) = 2}) = 3.
%e A305656 24 is a term because tau(24) = 8, omega(24) = 2, and #({phi(x) = 24}) = 10.
%p A305656 filter:= proc(n) uses numtheory; tau(n)+nops(factorset(n)) = nops(invphi(n)) end proc:
%p A305656 select(filter, [seq(i,i=2..10^5,2)]); # _Robert Israel_, Oct 28 2021
%t A305656 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 A305656 (PARI) isok(m) = numdiv(m) + omega(m) == #invphi(m); \\ _Michel Marcus_, Jun 08 2018
%Y A305656 Cf. A000005, A001222, A163523.
%Y A305656 Cf. A000010, A058277, A014197.
%Y A305656 Cf. A000079.
%K A305656 nonn
%O A305656 1,1
%A A305656 _Torlach Rush_, Jun 07 2018
%E A305656 More terms from _Michel Marcus_, Jun 08 2018