A326835 -id:A326835 - cp's OEIS frontend

A348198 Terms of A326835 having more divisors than any smaller term.

1, 3, 9, 15, 45, 105, 225, 405, 495, 1155, 3675, 4455, 8085, 19635, 62475, 75735, 137445, 373065, 1187025, 1741905, 2611455, 8580495, 27301575, 50515245, 60063465, 248834355, 1021078905, 2374216515, 2822982855, 11695214685, 47990708535

Offset: 1

Amiram Eldar, Oct 06 2021

All the terms are odd since all the terms of A326835 are odd (as phi(1) = phi(2) = 1).

The corresponding numbers of divisors are 1, 2, 3, 4, 6, 8, 9, 10, 12, 16, 18, 20, 24, 32, 36, 40, 48, 64, 72, 80, 96, 128, 144, 160, 192, 256, 288, 320, 384, 512, 576, ...

			The sequence A326835 begins with 1, 3, 5, 7, 9, 11, 13 and 15. The number of divisors of these terms are 1, 2, 2, 2, 3, 2, 2 and 4, respectively. The record values, 1, 2, 3 and 4, occur at 1, 3, 9 and 15, the first 4 terms of this sequence.

Cf. A326835, A328858, A348199.

q[n_] := Length @ Union[EulerPhi /@ (d = Divisors[n])] == Length[d]; dm = 0; s = {}; Do[If[q[n], d = DivisorSigma[0, n]; If[d > dm, dm = d; AppendTo[s, n]]], {n, 1, 10^6, 2}]; s

A348199 a(n) is the least term of A326835 whose number of divisors is n.

Original entry on oeis.org

1, 3, 9, 15, 81, 45, 729, 105, 225, 405, 59049, 495, 531441, 3645, 2025, 1155, 43046721, 3675, 387420489, 4455, 18225, 295245, 31381059609, 8085, 50625, 2657205, 65025, 40095, 22876792454961, 34425, 205891132094649, 19635, 1476225, 215233605, 455625, 62475

Offset: 1

Amiram Eldar, Oct 06 2021

nonn

First differs from A038547 at n = 12.
All the terms are odd since all the terms of A326835 are odd (as phi(1) = phi(2) = 1).
a(n) exists for any n since 3^(n-1) is a term of A326835 which has n divisors.

Cf. A038547, A326835, A328858, A348198.

seq[m_] := Module[{p = Select[Range[m], PrimeQ], s = Table[0, {m}], c, nd, ndd}, s[[p]] = 3^(p - 1); c = Length[p]; n = 1; While[c < m, nd = DivisorSigma[0, n]; If[nd <= m && s[[nd]] == 0, ndd = Length@Union[EulerPhi /@ Divisors[n]]; If[ndd == nd, c++; s[[nd]] = n]]; n++]; s]; seq[30]

a(n) <= 3^(n-1), with equality if n is prime.

a(n) >= A038547(n).

A348004 Numbers whose unitary divisors have distinct values of the unitary totient function uphi (A047994).

Original entry on oeis.org

1, 3, 4, 5, 7, 8, 9, 11, 12, 13, 15, 16, 17, 19, 20, 21, 23, 24, 25, 27, 28, 29, 31, 32, 33, 35, 36, 37, 39, 40, 41, 43, 44, 45, 47, 48, 49, 51, 52, 53, 55, 56, 57, 59, 60, 61, 63, 64, 65, 67, 68, 69, 71, 72, 73, 75, 76, 77, 79, 80, 81, 83, 85, 87, 88, 89, 91

Offset: 1

Amiram Eldar, Sep 23 2021

nonn

First differs from A042965 \ {0} at n=63, and from A122906 at n=53.
Since Sum_{d|k, gcd(d,k/d)=1} uphi(d) = k, these are numbers k such that the set {uphi(d) | d|k, gcd(d,k/d)=1} is a partition of k into distinct parts.
Includes all the odd prime powers (A061345), since an odd prime power p^e has 2 unitary divisors, 1 and p^e, whose uphi values are 1 and p^e - 1. It also includes all the powers of 2, except for 2 (A151821).
If k is a term, then all the unitary divisors of k are also terms.
The number of terms not exceeding 10^k for k = 1, 2, ... are 7, 74, 741, 7386, 73798, 737570, 7374534, 73740561, 737389031, 7373830133, ... Apparently, this sequence has an asymptotic density 0.73738...

			4 is a term since it has 2 unitary divisors, 1 and 4, and uphi(1) = 1 != uphi(4) = 3.
12 is a term since the uphi values of its unitary divisors, {1, 3, 4, 12}, are distinct: {1, 2, 3, 6}.

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

The unitary version of A326835.

Cf. A034444, A042965, A047994, A061345, A151821, A122906, A348001.

f[p_, e_] := p^e - 1; uphi[1] = 1; uphi[n_] := Times @@ f @@@ FactorInteger[n]; q[n_] := Length @ Union[uphi /@ (d = Select[Divisors[n], CoprimeQ[#, n/#] &])] == Length[d]; Select[Range[100], q]

from math import prod
from sympy.ntheory.factor_ import udivisors, factorint
A348004_list = []
for n in range(1,10**3):
    pset = set()
    for d in udivisors(n,generator=True):
        u = prod(p**e-1 for p, e in factorint(d).items())
        if u in pset:
            break
        pset.add(u)
    else:
        A348004_list.append(n) # Chai Wah Wu, Sep 24 2021

Numbers k such that A348001(k) = A034444(k).

A348158 a(n) is the sum of the distinct values obtained when the Euler totient function is applied to the divisors of n.

Original entry on oeis.org

1, 1, 3, 3, 5, 3, 7, 7, 9, 5, 11, 7, 13, 7, 15, 15, 17, 9, 19, 15, 21, 11, 23, 15, 25, 13, 27, 21, 29, 15, 31, 31, 33, 17, 35, 25, 37, 19, 39, 31, 41, 21, 43, 33, 45, 23, 47, 31, 49, 25, 51, 39, 53, 27, 55, 49, 57, 29, 59, 31, 61, 31, 57, 63, 65, 33, 67, 51, 69

Offset: 1

Amiram Eldar, Oct 03 2021

nonn

The sum of the distinct values of the n-th row of A102190.

Apparently, all the terms are odd.

			The divisors of 12 are {1, 2, 3, 4, 6, 12} and their phi values are {1, 1, 2, 2, 2, 4}. The set of distinct values is {1, 2, 4} whose sum is 7. Therefore, a(12) = 7.

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

Cf. A000010, A102190, A319696, A326835.

with(numtheory):
a:= n-> add(i, i=map(phi, divisors(n))):
seq(a(n), n=1..69);  # Alois P. Heinz, Nov 15 2021

a[n_] := Plus @@ DeleteDuplicates @ Map[EulerPhi, Divisors[n]]; Array[a, 100]

a(n) = vecsum(Set(apply(eulerphi, divisors(n)))); \\ Michel Marcus, Oct 04 2021

from sympy import totient, divisors
def A348158(n): return sum(set(map(totient,divisors(n,generator=True)))) # Chai Wah Wu, Nov 15 2021

a(n) <= n, with equality if and only if n is in A326835.

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

Original entry on oeis.org

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;

A348213 a(n) is the number of iterations that n requires to reach a fixed point under the map x -> A348158(x).

Original entry on oeis.org

0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 2, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0

Offset: 1

Amiram Eldar, Oct 07 2021

nonn

a(n) first differs from 1-A000035(n) at n = 63.
The number of iterations is finite for all n since A348158(n) <= n.
The fixed points are terms of A326835, so a(n) = 0 if and only if n is a term of A326835.

			a(1) = 0 since 1 is in A326835.
a(2) = 1 since there is one iteration of the map x -> A348158(x) starting from 2: 2 -> 1.
a(64) = 2 since there are 2 iterations of the map x -> A348158(x) starting from 64: 64 -> 63 -> 57.

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

Cf. A000035, A003434, A326835, A348158.

f[n_] := Plus @@ DeleteDuplicates @ Map[EulerPhi, Divisors[n]]; a[n_] := -2 + Length @ FixedPointList[f, n]; Array[a, 100]

from sympy import totient, divisors
def A348213(n):
    c, k = 0, n
    m = sum(set(map(totient,divisors(k,generator=True))))
    while m != k:
        k = m
        m = sum(set(map(totient,divisors(k,generator=True))))
        c += 1
    return c # Chai Wah Wu, Nov 15 2021

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);

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);

A359417 Phi-practical numbers (A260653) whose divisors have distinct values of the Euler totient function (A000010).

Original entry on oeis.org

1, 3, 15, 105, 165, 195, 255, 495, 525, 735, 975, 1155, 1485, 1785, 1815, 1995, 2145, 2415, 2535, 2625, 2805, 3045, 3135, 3255, 3315, 3675, 3705, 3795, 3885, 4305, 4455, 4485, 4515, 4785, 4845, 4875, 4935, 5115, 5145, 5445, 5565, 5655, 5865, 6045, 6105, 6195, 6405

Offset: 1

Amiram Eldar, Dec 31 2022

nonn

A phi-practical number k is a number k such that each number in the range 1..k is a subsum of a the multiset {phi(d) : d | k}. This sequence is restricted to cases in which all the values in this multiset are distinct.

Are all the terms above 3 divisible by 5?

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

Intersection of A260653 and A326835.

Cf. A000010.

phiPracticalQ[n_] := If[n<1, False, If[n==1, True, (lst = Sort @ EulerPhi @ Divisors[n]; ok = True; Do[If[lst[[m]]>Sum[lst[[l]], {l, 1, m-1}]+1, (ok=False; Break[])], {m, 1, Length[lst]}]; ok)]]; (* Frank M Jackson's code at A260653 *)
Select[Range[40000], UnsameQ @@ EulerPhi[Divisors[#]] && phiPracticalQ[#]  &]

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A348198 Terms of A326835 having more divisors than any smaller term.

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A348199 a(n) is the least term of A326835 whose number of divisors is n.

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A348004 Numbers whose unitary divisors have distinct values of the unitary totient function uphi (A047994).

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A348158 a(n) is the sum of the distinct values obtained when the Euler totient function is applied to the divisors of n.

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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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A348213 a(n) is the number of iterations that n requires to reach a fixed point under the map x -> A348158(x).

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

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