A373531 a(n) is the maximum number of divisors of n with an equal value of the Euler totient function (A000010).

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

1,2

Comments

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... .

Links

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

Formula

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.

Example

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.

Mathematica

a[n_] := Max[Tally[EulerPhi[Divisors[n]]][[;; , 2]]]; Array[a, 100]

Prog

(PARI) a(n) = vecmax(matreduce(apply(x->eulerphi(x), divisors(n)))[ , 2]);
(Python)
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

Crossrefs

Cf. A000010, A102190, A319696, A326835, A359563, A359565, A373532.

Sequence in context: A160981 A058656 A184221 * A226898 A338653 A033111

Adjacent sequences: A373528 A373529 A373530 * A373532 A373533 A373534

Keyword

nonn,easy

Author

Amiram Eldar, Jun 08 2024

Status

approved