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

2, 4, 8, 16, 24, 32, 64, 128, 256, 320, 512, 1024, 2048, 3712, 4096, 7168, 8192, 10512, 16192, 16384, 32768, 33024, 37888, 41728, 49280, 51552, 54528, 57280, 62592, 65536, 66432, 67968, 68832, 69792, 81600, 84352, 87696, 91968, 92016, 93888, 94720, 124128, 129888, 131072

Offset: 1

Author

Torlach Rush, Jun 07 2018

Keywords

• nonn

Comments

All even terms of A000079 are contained in this sequence.

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

Examples

			2 is a term because tau(2) = 2, omega(2) = 1, and #({phi(x) = 2}) = 3.
24 is a term because tau(24) = 8, omega(24) = 2, and #({phi(x) = 24}) = 10.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..1000
• Max Alekseyev, PARI scripts for various problems (see invphi.gp there).

Crossrefs

Cf. A000005, A001222, A163523.

Cf. A000010, A058277, A014197.

Cf. A000079.

Programs

Maple

filter:= proc(n) uses numtheory; tau(n)+nops(factorset(n)) = nops(invphi(n)) end proc:
select(filter, [seq(i,i=2..10^5,2)]); # Robert Israel, Oct 28 2021

Mathematica

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 *)

PARI

isok(m) = numdiv(m) + omega(m) == #invphi(m); \\ Michel Marcus, Jun 08 2018

Formula

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

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

Extensions

More terms from Michel Marcus, Jun 08 2018