A359565 Numbers that have at least three divisors with the same value of the Euler totient function (A000010).

12, 24, 36, 40, 48, 60, 72, 80, 84, 96, 108, 120, 126, 132, 144, 156, 160, 168, 180, 192, 200, 204, 216, 228, 240, 252, 264, 276, 280, 288, 300, 312, 320, 324, 336, 348, 360, 364, 372, 378, 384, 396, 400, 408, 420, 432, 440, 444, 456, 468, 480, 492, 504, 516, 520

Offset: 1

Author

Amiram Eldar, Jan 06 2023

Keywords

• nonn

Comments

The least odd term is a(392) = 3591, the least term that is coprime to 6 is a(34211) = 305515, and the least term that is coprime to 30 is a(158487) = 1413797.

If k is a term then all the multiples of k are terms. The primitive terms are in A359566.

The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 0, 10, 108, 1104, 11181, 112092, 1121784, 11221475, 112227492, 1122320814, ... . Apparently, the asymptotic density of this sequence exists and equals 0.1122... .

Examples

			12 is a term since its has 3 divisors, 3, 4 and 6, with the same value of the Euler totient function, 2.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000010, A326835, A359563, A359564, A359566.

Programs

Mathematica

Select[Range[1, 10^5, 2], Max[Tally[EulerPhi[Divisors[#]]][[;; , 2]]] > 2 &]

PARI

is(k) = vecmax(matreduce(apply(x->eulerphi(x), divisors(k)))[,2]) > 2;