decompwlj 3D

Numbers m for which the number of divisors, tau(m), is divisible by 3

A059269 on the OEIS · family divisor functions

Weight–level plate of Numbers m for which the number of divisors, tau(m), is divisible by 3
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 viewerA059269 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,998
Level class, k > L16,878 · 16.88 %
Weight class, k ≤ L83,120 · 83.12 %
Ties, k = L22
On the level line L = 15,071
Forced level, l ≤ d²4
Range of a(n)4 … 371,538
Range of the jump d1 … 8
Largest weight k, level L371,311, 185,750

8 different gaps occur, from 1 to 8; the level share is 16.88 %; L = 1 holds 30 % 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.