A348158 -id:A348158 - cp's OEIS frontend

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

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

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.

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

A348160 Odd numbers not appearing in A348158.

Original entry on oeis.org

189, 273, 315, 513, 567, 585, 825, 945, 1071, 1323, 1365, 1539, 1575, 1701, 1755, 1911, 2079, 2205, 2255, 2457, 2565, 2835, 3003, 3069, 3075, 3213, 3465, 3549, 3591, 3969, 4125, 4329, 4347, 4617, 4641, 4725, 4995, 5103, 5187, 5265, 5425, 5481, 5733, 5775, 5859, 5985

Offset: 1

Amiram Eldar, Oct 03 2021

nonn

a(19) = 2255 is the least term that is not divisible by 3.

Cf. A348158.

Similar sequences: A005114, A005277.

f[n_] := Plus @@ DeleteDuplicates @ Map[EulerPhi, Divisors[n]]; m = 6000; Complement[Range[1, m, 2], Array[f, m^2]]

A348173 a(n) is the sum of the distinct values obtained when the unitary totient function is applied to the unitary divisors of n.

Original entry on oeis.org

1, 1, 3, 4, 5, 3, 7, 8, 9, 5, 11, 12, 13, 7, 15, 16, 17, 9, 19, 20, 21, 11, 23, 24, 25, 13, 27, 28, 29, 15, 31, 32, 33, 17, 35, 36, 37, 19, 39, 40, 41, 21, 43, 44, 45, 23, 47, 48, 49, 25, 51, 52, 53, 27, 55, 56, 57, 29, 59, 60, 61, 31, 63, 64, 65, 33, 67, 68, 69, 35, 71, 72, 73, 37, 75, 76, 77, 39, 79, 80, 81, 41, 83, 78

Offset: 1

Amiram Eldar, Oct 04 2021

nonn

First differs from A283971 at n = 84.

			The unitary divisors of 18 are {1, 2, 9, 18} and their uphi values are {1, 1, 8, 8}. The set of distinct values is {1, 8} whose sum is 9. Therefore, a(18) = 9.

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

The unitary version of A348158.

Cf. A047994, A077610, A283971, A348001, A348004.

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

a(n) <= n, with equality if and only if n is in A348004.

A348214 a(n) is the least number k such that A348213(k) = n, or -1 if no such number exists.

Original entry on oeis.org

1, 2, 64, 50624, 235053, 15800785, 36903321, 4038974856

Offset: 0

Amiram Eldar, Oct 07 2021

			n The n iterations of a(n) under the map x -> A348158(x)
- --------------------------------------------------------------------------------
0 1
1 2 -> 1
2 64 -> 63 -> 57
3 50624 -> 49833 -> 49155 -> 48819
4 235053 -> 231363 -> 223245 -> 222885 -> 210693
5 15800785 -> 15775305 -> 15763125 -> 15761925 -> 15208875 -> 14889335
6 36903321 -> 36323991 -> 35049465 -> 34992945 -> 33078801 -> 32940117 -> 29802963
7 4038974856 -> 2855346375 -> 2854284615 -> 2556863361 -> 2549117805 -> 2536180173 -> 2447191395 -> 2445883515

Cf. A007755, A348158, A348213.

f[n_] := Plus @@ DeleteDuplicates @ Map[EulerPhi, Divisors[n]]; s[n_] := -2 + Length @ FixedPointList[f, n]; seq[m_, lim_] := Module[{t = Table[0, {m}], c = 0, n = 1}, While[c < m && n < lim, i = s[n] + 1; If[i <= m && t[[i]] == 0, c++; t[[i]] = n]; n++]; TakeWhile[t, # > 0 &]]; seq[5, 10^6]

A348216 Numbers k such that A348215(k) = k.

Original entry on oeis.org

120, 1320, 2760, 3480, 3720, 4920, 5160, 5640, 6360, 7080, 7320, 8040, 8520, 8760, 9480, 9960, 10680, 11640, 12120, 12360, 12840, 13080, 13560, 14520, 15240, 15720, 16440, 16680, 17880, 18120, 18840, 19560, 20040, 20760, 21480, 21720, 22920, 23160, 23640, 23880

Offset: 1

Amiram Eldar, Oct 07 2021

nonn

Are there odd terms in this sequence? There are none below 10^8.

			120 is a term since the iterations of the map x -> A348158(x) starting from 120 are 120 -> 63 -> 57 and A348215(120) = 57 + 63 = 120.

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

Cf. A082897, A348158, A348213, A348214, A348215.

f[n_] := Plus @@ DeleteDuplicates @ Map[EulerPhi, Divisors[n]]; s[n_] := Plus @@ Most @ FixedPointList[f, n] - n; Select[Range[24000], s[#] == # &]

A363320 a(n) is the product of the frequencies of the distinct values obtained when the Euler totient function is applied to the divisors of n.

Original entry on oeis.org

1, 2, 1, 2, 1, 4, 1, 2, 1, 4, 1, 6, 1, 4, 1, 2, 1, 8, 1, 4, 1, 4, 1, 12, 1, 4, 1, 4, 1, 16, 1, 2, 1, 4, 1, 12, 1, 4, 1, 6, 1, 16, 1, 4, 1, 4, 1, 24, 1, 8, 1, 4, 1, 16, 1, 4, 1, 4, 1, 54, 1, 4, 2, 2, 1, 16, 1, 4, 1, 16, 1, 24, 1, 4, 1, 4, 1, 16, 1, 12, 1, 4, 1, 36, 1, 4, 1, 4, 1, 64

Offset: 1

Juri-Stepan Gerasimov, May 27 2023

nonn

The product of the multiplicities of distinct values of the n-th row of A102190.

			The divisors of 12 are {1, 2, 3, 4, 6, 12} and their phi values are {1, 1, 2, 2, 2, 4} whose sum is also 12. The set of distinct values are {1, 2, 4} which occur with multiplicities {2, 3, 1} respectively. Therefore, a(12) = 2*3*1 = 6.

Cf. A000010, A102190, A348158.

a[n_] := Times @@ Tally[EulerPhi[Divisors[n]]][[;; , 2]]; Array[a, 100] (* Amiram Eldar, May 27 2023 *)

a(n)=my(f=vector(n)); fordiv(n,d,f[eulerphi(d)]++); vecprod([t | t<-f, t>0]) \\ Andrew Howroyd, May 27 2023

cp's OEIS Frontend

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

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

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A348160 Odd numbers not appearing in A348158.

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A348173 a(n) is the sum of the distinct values obtained when the unitary totient function is applied to the unitary divisors of n.

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A348214 a(n) is the least number k such that A348213(k) = n, or -1 if no such number exists.

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A348216 Numbers k such that A348215(k) = k.

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A363320 a(n) is the product of the frequencies of the distinct values obtained when the Euler totient function is applied to the divisors of n.

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