A297475 Numbers n such that phi(x) = n for more than one value of x, and the smallest such x divides the largest.

1, 2, 8, 10, 22, 28, 30, 44, 46, 52, 54, 56, 58, 66, 70, 78, 82, 92, 102, 104, 106, 110, 116, 126, 128, 130, 136, 138, 140, 148, 150, 164, 166, 172, 178, 190, 196, 198, 204, 210, 212, 222, 226, 228, 238, 250, 260, 262, 268, 270, 282, 292, 294, 296, 306, 310, 316, 330, 332, 342, 344, 346, 356, 358, 366, 368, 372

Offset

1,2

Comments

The larger endpoint is always twice the value of the smaller endpoint.

Conjecture 1: The number of solutions, excluding endpoints is always 0, or an odd number. (known to n = 2 * 10^5)

Conjecture 2: If both endpoints are divisible by 5, then the number of solutions (excluding terms of A007366) is of the form 4k + 1. (known to n = 2 * 10^5)

A007366 is contained in this sequence and the number of solutions, excluding endpoints is always 0.

Terms of this sequence are totients with a single odd totient inverse.

Links

Michel Marcus, Table of n, a(n) for n = 1..2441

Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Formula

2 = max({phi^-1(n)}) / min({phi^-1(n)}).

0 = A006511(n) mod A002181(n).

Example

2 is in the sequence because {phi^-1(2)} = {3,4,6}, and 2 = 6 / 3.
8 is in the sequence because {phi^-1(8)} = {15,...,30}, and 2 = 30 / 15.
10 is in the sequence because {phi^-1(10)} = {11,22}, and 2 = 22 / 11.

Mathematica

With[{nn = 67}, Take[#, nn] &@ Keys@ Select[KeySort@ PositionIndex@ Array[EulerPhi, nn^2], IntegerQ[#2/#1] & @@ {First@ #, Last@ #} &]] (* Michael De Vlieger, Dec 31 2017 *)

Prog

(PARI) isok(n) = my(vx = invphi(n)); (#vx > 1) && ((vecmax(vx) % vecmin(vx)) == 0); \\ Michel Marcus, Jul 18 2018

Crossrefs

Cf. A002181, A002202, A006511, A007366, A032447.

Sequence in context: A165593 A127219 A122208 * A106358 A209449 A002510

Adjacent sequences: A297472 A297473 A297474 * A297476 A297477 A297478

Keyword

nonn,changed

Author

Torlach Rush, Dec 30 2017

Status

approved