decompwlj 3D

Niven (or Harshad, or harshad) numbers: numbers that are divisible by the sum of their digits

A005349 on the OEIS · family digit rule · also known as Harshad numbers

Weight–level plate of Niven (or Harshad, or harshad) numbers: numbers that are divisible by the sum of their digits
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 viewerA005349 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,997
Level class, k > L20,334 · 20.33 %
Weight class, k ≤ L79,663 · 79.67 %
Ties, k = L4
On the level line L = 12,956
Forced level, l ≤ d²3
Range of a(n)1 … 1,033,781
Range of the jump d1 … 88
Largest weight k, level L1,033,661, 516,683

n divisible by its digit sum. Density by digit sum is 9:3:1 according to gcd(s,9), but that structure does not reach the plane: the weight fence is flat and the level share varies only between 11 and 24 %.

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.