decompwlj 3D

Non-Niven (or non-Harshad) numbers: numbers which are not a multiple of the sum of their digits

A065877 on the OEIS · family digit rule · also known as Non-Harshad numbers

Weight–level plate of Non-Niven (or non-Harshad) numbers: numbers which are not a multiple of 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 viewerA065877 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)100,000
Level class, k > L8,251 · 8.25 %
Weight class, k ≤ L91,749 · 91.75 %
Ties, k = L63
On the level line L = 18,163
Forced level, l ≤ d²0
Range of a(n)11 … 113,927
Range of the jump d1 … 5
Largest weight k, level L113,899, 56,961

Not divisible by the digit sum: the complement of the Harshad numbers. The level share is 8.25 %; L = 1 holds 99 % 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.