A359566 -id:A359566 - cp's OEIS frontend

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

Amiram Eldar, Jan 06 2023

nonn

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.

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

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

Complement of A326835 within the odd numbers.

Cf. A000010, A359564, A359565, A359566.

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 *)

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

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

Original entry on oeis.org

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

Amiram Eldar, Jan 06 2023

nonn

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... .

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

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

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

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

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

A359564 Primitive terms of A359563: terms of A359563 with no proper divisor in A359563.

Original entry on oeis.org

63, 273, 513, 585, 825, 2107, 2109, 2255, 3069, 3075, 4329, 4697, 4995, 5425, 5673, 6039, 6643, 6935, 6975, 7105, 7161, 8103, 8349, 8541, 8645, 9855, 10235, 11543, 12625, 13725, 13869, 14497, 14841, 16385, 18639, 18915, 19825, 22165, 25025, 26169, 26533, 30225

Offset: 1

Amiram Eldar, Jan 06 2023

nonn

Odd numbers that are not in A326835 but all of their proper divisors are in it.

If k is a term then all the odd multiples of k are terms of A359563.

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

Cf. A326835, A359563, A359565, A359566.

q[n_] := !UnsameQ @@ EulerPhi[Divisors[n]]; primQ[n_] := q[n] && AllTrue[Divisors[n], # == n || !q[#] &]; Select[Range[1, 30000, 2], primQ]

is1(k) = k>1 && k%2 && numdiv(k) > #Set(apply(x->eulerphi(x), divisors(k)));
is(k) = fordiv(k, d, if(is1(d), return(d==k))); return(0);

cp's OEIS Frontend

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

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

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PARI

A359564 Primitive terms of A359563: terms of A359563 with no proper divisor in A359563.

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Mathematica

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