decompwlj 3D

a(n) = a(n-1) + sum of digits of n

A037123 on the OEIS · family summatory

Weight–level plate of a(n) = a(n-1) + sum of digits of n
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 viewerA037123 on the OEIS
Terms100,000 (n = 0 … 99,999)
Decomposable (a > 2d)99,995
Level class, k > L28,313 · 28.31 %
Weight class, k ≤ L71,682 · 71.69 %
Ties, k = L0
On the level line L = 17,866
Forced level, l ≤ d²6
Range of a(n)0 … 2,250,000
Range of the jump d1 … 45
Largest weight k, level L2,249,867, 152,318

45 different gaps occur, from 1 to 45; the level share is 28.31 %; there are no ties.

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.