A326835 -id:A326835 - cp's OEIS frontend

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.

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

A348215 a(n) is the sum of the iterated A348158 starting from n until a fixed point is reached.

Original entry on oeis.org

0, 1, 0, 3, 0, 3, 0, 7, 0, 5, 0, 7, 0, 7, 0, 15, 0, 9, 0, 15, 0, 11, 0, 15, 0, 13, 0, 21, 0, 15, 0, 31, 0, 17, 0, 25, 0, 19, 0, 31, 0, 21, 0, 33, 0, 23, 0, 31, 0, 25, 0, 39, 0, 27, 0, 49, 0, 29, 0, 31, 0, 31, 57, 120, 0, 33, 0, 51, 0, 35, 0, 57, 0, 37, 0, 57

Offset: 1

Amiram Eldar, Oct 07 2021

nonn

The first odd number k with a(k) > 0 is k = 63.

			a(4) = 3 since the iterations of the map x -> A348158(x) starting from 4 are 4 -> 3.
a(64) = 120 since the iterations of the map x -> A348158(x) starting from 64 are 64 -> 63 -> 57, and 63 + 57 = 120.

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

Cf. A092693, A326835, A348158, A348213, A348214.

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

a(n) = 0 if and only if n is in A326835.

a(2*n) > 0 for all n.

A361924 Numbers whose infinitary divisors have distinct values of the infinitary totient function iphi (A091732).

Original entry on oeis.org

1, 3, 4, 5, 7, 9, 11, 12, 13, 15, 16, 17, 19, 20, 21, 23, 25, 27, 28, 29, 31, 33, 35, 36, 37, 39, 41, 43, 44, 45, 47, 48, 49, 51, 52, 53, 55, 57, 59, 60, 61, 63, 64, 65, 67, 68, 69, 71, 73, 75, 76, 77, 79, 80, 81, 83, 85, 87, 89, 91, 92, 93, 95, 97, 99, 100, 101

Offset: 1

Amiram Eldar, Mar 30 2023

nonn

First differs from A003159 at n = 57.
Numbers k such that A361923(k) = A037445(k).
Since Sum_{d infinitary divisor of k} iphi(d) = k, these are numbers k such that the multiset {iphi(d) | d infinitary divisor of k} is a partition of k into distinct parts.
Includes all the odd prime powers (A061345) and all the powers of 4 (A000302).
The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 6, 66, 651, 6497, 64894, 648641, 6485605, 64851632, 648506213, 6485025363, ... . Apparently, this sequence has an asymptotic density 0.6485...

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

Cf. A000302, A003159, A037445, A061345, A077609, A091732, A361923.

Similar sequences: A326835, A348004.

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};
q[n_] := Length @ Union[iphi /@ (d = idivs[n])] == Length[d]; Select[Range[100], q]

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
is(k) = {my(d = idivs(k)); #Set(apply(x->iphi(x), d)) == #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.

A348159 Indices k of records of low value in the ratios A348158(k)/k.

Original entry on oeis.org

1, 2, 126, 1638, 2394, 8190, 139230, 155610, 2645370, 5757570, 97878690, 420302610, 1963331370, 7145144370

Offset: 1

Amiram Eldar, Oct 03 2021

The maximal possible value of the ratio A348158(k)/k is 1 which occurs at the terms of A326835.
The rounded values of the corresponding records are 1, 0.5, 0.452, 0.445, 0.437, 0.424, 0.420, 0.409, 0.404, 0.398, 0.3933, 0.3927, 0.3885, 0.3879, ...
a(15) <= 33376633290. - David A. Corneth, Oct 04 2021

Cf. A326835, A328859, A348158.

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

f(n) = vecsum(Set(apply(eulerphi, divisors(n)))); \\ A348158
lista(nn) = {my(r=oo, x); for (i=1, nn, if ((x=f(i)/i) < r, print1(i, ", "); r = x););} \\ Michel Marcus, Oct 04 2021

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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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A348215 a(n) is the sum of the iterated A348158 starting from n until a fixed point is reached.

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A361924 Numbers whose infinitary divisors have distinct values of the infinitary totient function iphi (A091732).

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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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A348159 Indices k of records of low value in the ratios A348158(k)/k.

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