A359563 Odd numbers that have at least two divisors with the same value of the Euler totient function (A000010).

63, 189, 273, 315, 441, 513, 567, 585, 693, 819, 825, 945, 1071, 1197, 1323, 1365, 1449, 1539, 1575, 1701, 1755, 1827, 1911, 1953, 2079, 2107, 2109, 2205, 2255, 2331, 2457, 2475, 2565, 2583, 2709, 2835, 2925, 2961, 3003, 3069, 3075, 3087, 3213, 3339, 3465, 3549

Offset: 1

Author

Amiram Eldar, Jan 06 2023

Keywords

• nonn

Comments

The even numbers are excluded from this sequence since every even number has this property: it is divisible by 1 and 2, and phi(1) = phi(2) = 1.

If k is a term then all the odd multiples of k are terms. The primitive terms are in A359564.

The numbers of terms below 10^k, for k = 1, 2, ..., are 0, 1, 12, 140, 1402, 14193, 142606, 1427749, 14283236, 142855925, ... . Apparently, the asymptotic density of this sequence exists and equals 0.01428... .

The least term that is not divisible by 3 is a(26) = 2107.

Examples

			63 is a term since it is odd, 7 and 9 are both divisors of 63, and phi(7) = phi(9) = 6.
		

Links

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

Crossrefs

Complement of A326835 within the odd numbers.

Cf. A000010, A359564, A359565, A359566.

Programs

Mathematica

Select[Range[1, 3500, 2], !UnsameQ @@ EulerPhi[Divisors[#]] &]
Select[Range[1,3601,2],Max[Tally[EulerPhi[Divisors[#]]][[;;,2]]]>1&] (* Harvey P. Dale, Mar 05 2025 *)

PARI

is(k) = k>1 && k%2 && numdiv(k) > #Set(apply(x->eulerphi(x), divisors(k)));