A359563 -id:A359563 - cp's OEIS frontend

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

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);

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;

A359566 Primitive terms of A359565: terms of A359565 with no proper divisor in A359565.

Original entry on oeis.org

12, 40, 126, 364, 544, 546, 1026, 1100, 1170, 1650, 2812, 3591, 3608, 4095, 4100, 4214, 4218, 4510, 6138, 6150, 7564, 8658, 9394, 9548, 9990, 10804, 10850, 11096, 11132, 11346, 11368, 12078, 13286, 13870, 13950, 14210, 14322, 16206, 16376, 16698, 17082, 17290

Offset: 1

Amiram Eldar, Jan 06 2023

nonn

If k is a term then all the positive multiples of k are terms of A359565.

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

Cf. A326835, A359563, A359564, A359565.

q[n_] := Max[Tally[EulerPhi[Divisors[n]]][[;; , 2]]] > 2; primQ[n_] := q[n] && AllTrue[Divisors[n], # == n || !q[#] &]; Select[Range[17000], primQ]

is1(k) = vecmax(matreduce(apply(x->eulerphi(x), divisors(k)))[,2]) > 2;
is(k) = fordiv(k, d, if(is1(d), return(d==k))); return(0);

A373527 Odd numbers k such that k and k+2 both have at least two divisors with the same value of the Euler totient function (A000010).

Original entry on oeis.org

2107, 11275, 42651, 68733, 90153, 99123, 123633, 213003, 226825, 242305, 262143, 272853, 292873, 295405, 308007, 313443, 376675, 376803, 378693, 390115, 427425, 471293, 473263, 524797, 525481, 556983, 579535, 591325, 618469, 638163, 663325, 669123, 699853, 731815

Offset: 1

Amiram Eldar, Jun 08 2024

nonn

Numbers k such that k and k+2 are both in A359563.

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

Subsequence of A359563.
A373528 is a subsequence.
Cf. A000010, A102190.

q[n_] := q[n] = UnsameQ @@ EulerPhi[Divisors[n]]; Select[Range[1, 10^6, 2], ! q[#] && ! q[# + 2] &]

is(k) = k>1 && k%2 && numdiv(k) > #Set(apply(x->eulerphi(x), divisors(k)));
lista(kmax) = {my(q1 = 0, q2); forstep(k = 3, kmax, 2, q2 = is(k); if(q1 && q2, print1(k-2, ", ")); q1 = q2);}

A373528 Odd numbers k such that k, k+2 and k+4 all have at least two divisors with the same value of the Euler totient function (A000010).

Original entry on oeis.org

4142435, 26196331, 77118741, 89690821, 102974571, 196054673, 201060275, 206568171, 277322153, 280039833, 401784953, 402492695, 415097613, 437290371, 515636303, 526721895, 534746581, 549806211, 575090395, 580329603, 625833871, 629588043, 702183625, 710983971, 716133481

Offset: 1

Amiram Eldar, Jun 08 2024

nonn

Numbers k such that k, k+2 and k+4 are all in A359563.

Amiram Eldar, Table of n, a(n) for n = 1..100

Subsequence of A359563 and A373527.

Cf. A000010, A102190.

q[n_] := !UnsameQ @@ EulerPhi[Divisors[n]]; seq[kmax_] := Module[{tri = q /@ {1, 3, 5}, s = {}, k = 7}, While[k < kmax, If[And @@ tri, AppendTo[s, k - 6]]; tri = Join[Rest[tri], {q[k]}]; k+=2]; s]; seq[3*10^7]

is(k) = k>1 && k%2 && numdiv(k) > #Set(apply(x->eulerphi(x), divisors(k)));
lista(kmax) = {my(q1 = 0, q2 = 0, q3); forstep(k = 5, kmax, 2, q3 = is(k); if(q1 && q2 && q3, print1(k-4, ", ")); q1 = q2; q2 = q3);}

A373531 a(n) is the maximum number of divisors of n with an equal value of the Euler totient function (A000010).

Original entry on oeis.org

1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 1

Offset: 1

Amiram Eldar, Jun 08 2024

The sums of the first 10^k terms, for k = 1, 2, ..., are 15, 161, 1641, 16554, 166029, 1662306, 16630535, 166335597, 1663473941, 16635216306, ... . Apparently, this sequence has an asymptotic mean 1.663... .

			a(2) = 2 since 2 has 2 divisors, 1 and 2, and phi(1) = phi(2) = 1.
a(12) = 3 since 3 of the divisors of 12 (3, 4 and 6) have the same value of phi: phi(3) = phi(4) = phi(6) = 2.

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

Cf. A000010, A102190, A319696, A326835, A359563, A359565, A373532.

a[n_] := Max[Tally[EulerPhi[Divisors[n]]][[;; , 2]]]; Array[a, 100]

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

from collections import Counter
from sympy import divisors, totient
def a(n):
    c = Counter(totient(d) for d in divisors(n, generator=True))
    return c.most_common(1)[0][1]
print([a(n) for n in range(1, 88)]) # Michael S. Branicky, Jun 08 2024

a(A326835(n)) = 1.
a(A359563(n)) >= 2.
a(A359565(n)) >= 3.
a(2*n) >= 2.
a(p) = 2 for an odd prime p.
a(m*n) >= a(n) for all m > 1.

cp's OEIS Frontend

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

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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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A359566 Primitive terms of A359565: terms of A359565 with no proper divisor in A359565.

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A373527 Odd numbers k such that k and k+2 both have at least two divisors with the same value of the Euler totient function (A000010).

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A373528 Odd numbers k such that k, k+2 and k+4 all have at least two divisors with the same value of the Euler totient function (A000010).

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A373531 a(n) is the maximum number of divisors of n with an equal value of the Euler totient function (A000010).

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