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

Author

Amiram Eldar, Oct 07 2021

Keywords

• nonn

Comments

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.

Examples

			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.
		

Links

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

Crossrefs

Cf. A000035, A003434, A326835, A348158.

Programs

Mathematica

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

Python

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