Numbers j such that the average of the divisors of j is an integer: sigma_0(j) divides sigma_1(j). Alternatively, numbers j such that tau(j) (A000005(j)) divides sigma(j) (A000203(j))

A003601 on the OEIS · family divisor functions · also known as Arithmetic numbers

Weight–level plate of Numbers j such that the average of the divisors of j is an integer: sigma_0(j) divides sigma_1(j). Alternatively, numbers j such that tau(j) (A000005(j)) divides sigma(j) (A000203(j)) — Click the plate to explore it in 3-D. Weight k across, level L up, both on log scales: blue in the weight class (k ≤ L), orange in the level class (k > L). The dashed diagonal is k = L.

Open in the 3-D viewer A003601 on the OEIS

Terms	100,000 (n = 1 … 100,000)
Decomposable (a > 2d)	99,997
Level class, k > L	11,933 · 11.93 %
Weight class, k ≤ L	88,064 · 88.07 %
Ties, k = L	64
On the level line L = 1	11,627
Forced level, l ≤ d²	0
Range of a(n)	1 … 125,587
Range of the jump d	1 … 6
Largest weight k, level L	125,551, 62,793

The mean of the divisors, sigma(n)/d(n), is an integer. The level share is 11.93 %; L = 1 holds 97 % of the level class.

The decomposition

Every term of a strictly increasing sequence is written a(n) = k(n)·L(n) + d(n): the jump d = a(n+1) − a(n), the weight k the least divisor of a − d greater than d, the level L = (a − d)/k. A term decomposes when a > 2d; it is in the level class when k > L and in the weight class when k ≤ L. See decompwlj.com and arXiv:0711.0865, or how it works, with worked examples and a live one.