decompwlj 3D

a(0) = 1, a(n) = sum of digits of all previous terms

A004207 on the OEIS · family digit rule · also known as Digit-addition trajectory

Weight–level plate of a(0) = 1, a(n) = sum of digits of all previous terms
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 viewerA004207 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,996
Level class, k > L14,194 · 14.19 %
Weight class, k ≤ L85,802 · 85.81 %
Ties, k = L1
On the level line L = 13
Forced level, l ≤ d²12
Range of a(n)1 … 2,609,882
Range of the jump d1 … 50
Largest weight k, level L289,937, 300,015

a(n) = a(n-1) + digitsum(a(n-1)), so 9 divides l at every term. Above a = 800 every level term lies on a line L in 9Z (a level term off 9Z needs l <= 3d^2, impossible past a = 10^4), and Lemma 3 (L <= d) caps the lines at the gap: with d <= 50 in this range only L = 9, 18, 27, 36 occur. The 12 level terms off 9Z are all below 800.

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.