A391374 Number of positive integers k such that k's arithmetic mean of divisors is at most n.

1, 3, 6, 8, 11, 14, 18, 20, 23, 26, 30, 35, 37, 41, 45, 47, 49, 55, 60, 62, 68, 69, 72, 79, 82, 84, 90, 94, 96, 102, 106, 111, 113, 115, 116, 124, 127, 134, 138, 141, 144, 153, 153, 156, 169, 171, 173, 178, 183, 184, 186, 190, 191, 199, 201, 208, 216, 216, 218, 227

Offset

1,2

Comments

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

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..60.

Formula

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

Example

For n=2, we need sigma(k)/tau(k) <= 2. The qualifying integers are 1, 2, 3 so a(2) = 3.
For n=3, we need sigma(k)/tau(k) <= 3. The additional qualifying integers are 4, 5, 6 so a(3) = 6.

Prog

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

Crossrefs

Cf. A000005, A000203, A391368.

Sequence in context: A362882 A265898 A133869 * A368642 A356395 A276573

Adjacent sequences: A391371 A391372 A391373 * A391375 A391376 A391377

Keyword

nonn

Author

Joe Anderson, Dec 07 2025

Status

approved