A296655 Numbers k such that phi(x) = k has a positive even number of solutions.

1, 4, 6, 10, 12, 16, 18, 22, 24, 28, 30, 36, 42, 46, 52, 54, 58, 64, 66, 70, 78, 80, 82, 84, 88, 100, 102, 106, 110, 112, 126, 130, 136, 138, 148, 150, 160, 162, 166, 168, 172, 176, 178, 180, 184, 190, 196, 198, 200, 208, 210, 222, 224, 226, 228, 232, 238, 250

Offset: 1

Author

Torlach Rush, Dec 17 2017

Keywords

• nonn

Comments

When the number of solutions is 2, the sum of Sum_{d|x} d*mu(d) is always 0.

A007366 is contained in this sequence because it selects terms with the smallest even number of inverses.

Examples

			1 is a term because phi(1) has two inverses, 1, and 2.
6 is a term because phi(6) has four inverses, 7, 9, 14, and 18.
10 is a term because phi(10) has two inverses, 11, and 22.
18 is a term because phi(18) has four inverses, 19, 27, 38, 54.
348 is a term because phi(348) has six inverses, 349, 413, 531, 698, 826, and 1062.
		

Crossrefs

Cf. A007366, A014197, A032446, A071386.

Programs

Mathematica

With[{nn = 1500}, TakeWhile[Union@ Select[KeyValueMap[{#1, Length@ #2} &, PositionIndex@ Array[EulerPhi, nn]], EvenQ@ Last@ # &][[All, 1]], # <= nn/6 &] ] (* Michael De Vlieger, Dec 20 2017 *)

Formula

0 = card({phi^-1(a(n))}) mod 2.

Extensions

Corrected and extended by Rémy Sigrist, Dec 19 2017