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URL: https://oeis.org/A391374

⇱ A391374 - OEIS


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A391374
Number of positive integers k such that k's arithmetic mean of divisors is at most n.
0
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.
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
KEYWORD
nonn
AUTHOR
Joe Anderson, Dec 07 2025
STATUS
approved