A391368 Number of positive integers k with an arithmetic mean of divisors equal to n.

1, 1, 2, 1, 0, 3, 2, 1, 3, 2, 0, 4, 1, 3, 4, 1, 0, 5, 2, 1, 5, 1, 0, 5, 0, 2, 4, 4, 0, 6, 2, 1, 1, 1, 1, 8, 1, 2, 3, 3, 0, 8, 0, 1, 9, 0, 0, 5, 3, 0, 2, 3, 0, 8, 1, 5, 5, 0, 0, 9, 1, 5, 10, 2, 3, 3, 0, 1, 1, 7, 0, 10, 0, 1, 2, 2, 0, 5, 1, 2, 5, 1, 0, 12, 0, 0

Offset

1,3

Comments

a(n) is the number of integers k such that sigma(k)/tau(k) = A000203(k)/A000005(k) = n.

Zeros occur if n is in A157847 (not in A157846).

a(n) is finite for all n since sigma(k) >= k+1 and tau(k) <= 2*sqrt(k) implies no qualifying k can exceed 4*n^2.

Links

Table of n, a(n) for n=1..86.

Formula

a(n) = |{k in Z+ : sigma(k) = n*tau(k)}|.

Example

For n=2, we need sigma(k)/tau(k) = 2. The only qualifying integer is 3 so a(2) = 1.
For n=6, we need sigma(k)/tau(k) = 6. The qualifying integers are 11, 14, 15 so a(6) = 3.

Prog

(Python)
from sympy import divisor_count, divisor_sigma
def a(n):
    count = 0
    for k in range(1, 4*n*n + 1):
        if divisor_sigma(k) == n * divisor_count(k): count += 1
    return count

Crossrefs

Cf. A000005, A000203, A162538, A157847, A157846.

Sequence in context: A389733 A391983 A255325 * A025654 A025648 A025655

Adjacent sequences: A391365 A391366 A391367 * A391369 A391370 A391371

Keyword

nonn

Author

Joe Anderson, Dec 07 2025

Extensions

More terms from Sean A. Irvine, Dec 14 2025

Status

approved