decompwlj 3D

Numbers divisible both by their individual digits and by the sum of their digits

A051004 on the OEIS · family digit rule

Weight–level plate of Numbers divisible both by their individual digits and 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 viewerA051004 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,995
Level class, k > L19,341 · 19.34 %
Weight class, k ≤ L80,654 · 80.66 %
Ties, k = L1
On the level line L = 19
Forced level, l ≤ d²1,636
Range of a(n)1 … 77,917,896
Range of the jump d1 … 11,111,652
Largest weight k, level L12,962,604, 9,047,460

1,074 different gaps occur, from 1 to 11,111,652; the level share is 19.34 %; 1.6 % of terms are forced level (l <= d^2).

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.