login
A359565
Numbers that have at least three divisors with the same value of the Euler totient function (A000010).
7
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
1,1
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... .
LINKS
EXAMPLE
12 is a term since its has 3 divisors, 3, 4 and 6, with the same value of the Euler totient function, 2.
MATHEMATICA
Select[Range[1, 10^5, 2], Max[Tally[EulerPhi[Divisors[#]]][[;; , 2]]] > 2 &]
PROG
(PARI) is(k) = vecmax(matreduce(apply(x->eulerphi(x), divisors(k)))[, 2]) > 2;
CROSSREFS
KEYWORD
nonn
AUTHOR
Amiram Eldar, Jan 06 2023
STATUS
approved