A319696 -id:A319696 - cp's OEIS frontend

A328858 Numbers with a record number of distinct values of the Euler totient function applied to their divisors (A319696).

1, 3, 8, 15, 32, 45, 72, 105, 144, 216, 288, 432, 792, 864, 1296, 1584, 1728, 2376, 2592, 3168, 4752, 9504, 14256, 19008, 28512, 38016, 54000, 57024, 85536, 108000, 114048, 162000, 171072, 216000, 218592, 228096, 324000, 342144, 437184, 465696, 648000, 655776

Offset: 1

Amiram Eldar, Oct 28 2019

nonn

The corresponding record values are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 16, 17, 18, 19, 20, 21, 22, 26, 32, 34, 38, 42, 44, 45, 50, 52, 54, 58, 59, 62, 63, 64, 66, 71, 72, 76, 78, 83, 84, ...

			The first 10 terms of A319696(k) are 1, 1, 2, 2, 2, 2, 2, 3, 3, 2. The record values 1, 2, and 3 are obtained at k = 1, 3, and 8. Therefore this sequence begins with 1, 3, 8.

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

Cf. A000010, A102190, A319696.

f[n_] := Length @ Union[EulerPhi /@ Divisors[n]]; fm = 0; s = {}; Do[f1 = f[n]; If[f1 > fm, fm = f1; AppendTo[s, n]], {n, 1, 660000}]; s

A328859 Indices k of records of low value in the ratios A319696(k)/A000005(k) between the number of distinct values of the Euler totient function applied to the divisors of k and the number of divisors of k.

Original entry on oeis.org

1, 2, 60, 120, 240, 480, 960, 1920, 3840, 4080, 8160, 16320, 32640, 65280, 130560, 261120, 522240, 1044480, 1485120, 2227680, 2970240, 4455360, 8910720, 17821440, 35642880, 42325920, 63488880, 69090840, 84651840, 126977760, 169303680, 253955520, 507911040, 761866560

Offset: 1

Amiram Eldar, Oct 28 2019

nonn

The maximal possible value of the ratio A319696(k)/A000005(k) is 1 which occurs at the terms of A326835.

The rounded values of the corresponding ratios are 1, 0.5, 0.417, 0.375, 0.35, 0.333, 0.321, 0.313, 0.306, 0.275, 0.25, 0.232, 0.219, 0.208, 0.2, 0.193, 0.188, 0.183, 0.179, 0.170, 0.168, 0.158, 0.148, 0.141, 0.135, 0.132, 0.130, 0.129, 0.122, 0.117, 0.115, 0.108, 0.102, 0.101, ...

Cf. A000010, A102190, A319696, A326835.

r[n_] := Length @ Union[EulerPhi /@ (d = Divisors[n])]/Length[d]; rm = 2; s = {}; Do[r1 = r[n]; If[r1 < rm, rm = r1; AppendTo[s, n]], {n, 1, 10^5}]; s

A326835 Numbers whose divisors have distinct values of the Euler totient function (A000010).

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 127

Offset: 1

Amiram Eldar, Oct 28 2019

nonn

Since Sum_{d|k} phi(d) = k, these are numbers k such that the set {phi(d) | d|k} is a partition of k into distinct parts.
Includes all the odd prime numbers, since an odd prime p has 2 divisors, 1 and p, whose phi values are 1 and p-1.
If k is a term, then all the divisors of k are also terms. If k is not a term, then all its multiples are not terms. The primitive terms of the complementary sequence are 2, 63, 273, 513, 585, 825, 2107, 2109, 2255, 3069, ....
In particular, all the terms are odd since 2 is not a term (phi(1) = phi(2)).
The number of terms below 10^k for k = 1, 2, ... are 5, 49, 488, 4860, 48598, 485807, 4857394, 48572251, 485716764, 4857144075, ...
Apparently the sequence has an asymptotic density of 0.4857...

			3 is a term since it has 2 divisors, 1 and 3, and phi(1) = 1 != phi(3) = 2.
15 is a term since the phi values of its divisors, {1, 3, 5, 15}, are distinct: {1, 2, 4, 8}.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000005, A000010, A032741, A102190, A319695, A319696.

filter:= proc(n) local D;
  D:=numtheory:-divisors(n);
  nops(D) = nops(map(numtheory:-phi,D))
end proc:
select(filter, [seq(i,i=1..200,2)]); # Robert Israel, Oct 29 2019

aQ[n_] := Length @ Union[EulerPhi /@ (d = Divisors[n])] == Length[d];  Select[Range[130], aQ]

isok(k) = #Set(apply(x->eulerphi(x), divisors(k))) == numdiv(k); \\ Michel Marcus, Oct 28 2019

Numbers k such that A319696(k) = A000005(k).
Numbers k such that A319695(k) = A032741(k).
Numbers k such that the k-th row of A102190 has distinct terms.

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.

A319695 Number of distinct values obtained when Euler phi (A000010) is applied to proper divisors of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 3, 3, 1, 3, 1, 3, 3, 2, 1, 3, 2, 2, 3, 3, 1, 4, 1, 4, 3, 2, 3, 4, 1, 2, 3, 4, 1, 4, 1, 3, 5, 2, 1, 4, 2, 3, 3, 3, 1, 4, 3, 5, 3, 2, 1, 4, 1, 2, 4, 5, 3, 4, 1, 3, 3, 4, 1, 6, 1, 2, 5, 3, 3, 4, 1, 5, 4, 2, 1, 5, 3, 2, 3, 5, 1, 6, 3, 3, 3, 2, 3, 5, 1, 3, 5, 5, 1, 4, 1, 5, 7

Offset: 1

Antti Karttunen, Oct 02 2018

nonn

			For n = 6, it has three proper divisors: 1, 2, 3, and applying A000010 to these gives 1, 1 and 2, with just two distinct values, thus a(6) = 2.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000010, A319696.

Cf. also A304793, A305611, A316555, A316556, A319685 for similarly constructed sequences.

A319695(n) = { my(m=Map(),s,k=0); fordiv(n,d,if((d

A348001 Number of distinct values obtained when the unitary totient function (A047994) is applied to the unitary divisors of n.

Original entry on oeis.org

1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 4, 2, 2, 2, 2, 4, 4, 2, 2, 4, 2, 2, 2, 4, 2, 4, 2, 2, 4, 2, 4, 4, 2, 2, 4, 4, 2, 4, 2, 4, 4, 2, 2, 4, 2, 2, 4, 4, 2, 2, 4, 4, 4, 2, 2, 8, 2, 2, 4, 2, 4, 4, 2, 4, 4, 4, 2, 4, 2, 2, 4, 4, 4, 4, 2, 4, 2, 2, 2, 7, 4, 2, 4

Offset: 1

Amiram Eldar, Sep 23 2021

nonn

			n = 6 has four unitary divisors: 1, 2, 3 and 6. Applying A047994 to these gives 1, 1, 2 and 2, with just 2 distinct values, thus a(6) = 2.
n = 12 has four unitary divisors: 1, 3, 4 and 12. Applying A047994 to these gives 4 distinct values, 1, 2, 3 and 6, thus a(12) = 4.

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

The unitary version of A319696.

Cf. A047994, A077610, A278568.

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

a(2^e) = 2 for e > 1.
a(p^e) = 2 for an odd prime p and e > 0.
a(n) >= omega(n), with equality if and only if n is in A278568.

A319686 Number of distinct values obtained when arithmetic derivative (A003415) is applied to the divisors of n.

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 2, 4, 3, 3, 2, 5, 2, 3, 3, 5, 2, 5, 2, 5, 3, 3, 2, 7, 3, 3, 4, 5, 2, 6, 2, 6, 3, 3, 3, 8, 2, 3, 3, 7, 2, 6, 2, 5, 5, 3, 2, 9, 3, 5, 3, 5, 2, 7, 3, 7, 3, 3, 2, 10, 2, 3, 5, 7, 3, 6, 2, 5, 3, 6, 2, 11, 2, 3, 5, 5, 3, 6, 2, 9, 5, 3, 2, 10, 3, 3, 3, 7, 2, 10, 3, 5, 3, 3, 3, 11, 2, 5, 5, 8, 2, 6, 2, 7, 6, 3, 2, 11, 2, 6, 3, 8, 2, 6, 3, 5, 5, 3, 3, 14

Offset: 1

Antti Karttunen, Oct 02 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

One more than A319685.
Cf. A003415.
Cf. also A184395, A319696.

d[0] = d[1] = 0; d[n_] := d[n] = n * Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); a[n_] := CountDistinct[d /@ Divisors[n]]; Array[a, 100] (* Amiram Eldar, Apr 17 2024 *)

A003415(n) = {my(fac); if(n<1, 0, fac=factor(n); sum(i=1, matsize(fac)[1], n*fac[i, 2]/fac[i, 1]))}; \\ From A003415
A319686(n) = { my(m=Map(),s,k=0); fordiv(n,d,if(!mapisdefined(m,s=A003415(d)), mapput(m,s,s); k++)); (k); };

a(n) = my(d = divisors(n)); for(i = 1, #d, d[i] = A003415(d[i])); #Set(d) \\ uses A003415 listed at Antti's programs. David A. Corneth, Oct 02 2018

a(n) = 1+A319685(n).

A361923 Number of distinct values obtained when the infinitary totient function (A091732) is applied to the infinitary divisors of n.

Original entry on oeis.org

1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 4, 2, 2, 2, 2, 4, 4, 2, 2, 4, 2, 2, 4, 4, 2, 4, 2, 2, 4, 2, 4, 4, 2, 2, 4, 4, 2, 4, 2, 4, 4, 2, 2, 4, 2, 2, 4, 4, 2, 4, 4, 4, 4, 2, 2, 8, 2, 2, 4, 4, 4, 4, 2, 4, 4, 4, 2, 4, 2, 2, 4, 4, 4, 4, 2, 4, 2, 2, 2, 7, 4, 2, 4

Offset: 1

Amiram Eldar, Mar 30 2023

nonn

First differs from A348001 at n = 27.

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

Cf. A077609, A091732.

Similar sequences: A319696, A348001.

f[p_, e_] := p^(2^(-1 + Position[Reverse@ IntegerDigits[e, 2], 1]));
iphi[1] = 1; iphi[n_] := Times @@ (Flatten@ (f @@@ FactorInteger[n]) - 1);
idivs[n_] := Sort@ Flatten@ Outer[Times, Sequence @@ (FactorInteger[n] /. {p_, m_Integer} :> p^Select[Range[0, m], BitOr[m, #] == m &])]; idivs[1] = {1};
a[n_] := Length @ Union[iphi /@ idivs[n]]; Array[a, 100]

iphi(n) = {my(f=factor(n), b); prod(i=1, #f~, b = binary(f[i, 2]); prod(k=1, #b, if(b[k], f[i, 1]^(2^(#b-k)) - 1, 1)))}
isidiv(d, f) = {if (d==1, return (1)); for (k=1, #f~, bne = binary(f[k, 2]); bde = binary(valuation(d, f[k, 1])); if (#bde < #bne, bde = concat(vector(#bne-#bde), bde)); for (j=1, #bne, if (! bne[j] && bde[j], return (0)); ); ); return (1); }
idivs(n) = {my(d = divisors(n), f = factor(n), idiv = []); for (k=1, #d, if(isidiv(d[k], f), idiv = concat(idiv, d[k])); ); idiv; } \\ Michel Marcus at A077609
a(n) = {my(d = idivs(n)); #Set(apply(x->iphi(x), d));}

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

A328858 Numbers with a record number of distinct values of the Euler totient function applied to their divisors (A319696).

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A328859 Indices k of records of low value in the ratios A319696(k)/A000005(k) between the number of distinct values of the Euler totient function applied to the divisors of k and the number of divisors of k.

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A326835 Numbers whose divisors have distinct values of the Euler totient function (A000010).

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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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A319695 Number of distinct values obtained when Euler phi (A000010) is applied to proper divisors of n.

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A348001 Number of distinct values obtained when the unitary totient function (A047994) is applied to the unitary divisors of n.

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A319686 Number of distinct values obtained when arithmetic derivative (A003415) is applied to the divisors of n.

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A361923 Number of distinct values obtained when the infinitary totient function (A091732) is applied to the infinitary divisors of n.

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