decompwlj 3D

Self numbers divisible by sum of their digits (or, self numbers which are also Harshad numbers)

A003219 on the OEIS · family digit rule

Weight–level plate of Self numbers divisible by sum of their digits (or, self numbers which are also Harshad numbers)
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 viewerA003219 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,994
Level class, k > L33,169 · 33.17 %
Weight class, k ≤ L66,825 · 66.83 %
Ties, k = L2
On the level line L = 12,884
Forced level, l ≤ d²131
Range of a(n)1 … 10,322,263
Range of the jump d2 … 810
Largest weight k, level L10,319,423, 3,389,576

216 different gaps occur, from 2 to 810; the level share is 33.17 %; 6 terms do not decompose.

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.