decompwlj 3D

a(n+1) = a(n) + sum of digits of a(n)^2, with a(1) = 1

A033298 on the OEIS · family digit rule

Weight–level plate of a(n+1) = a(n) + sum of digits of a(n)^2, with a(1) = 1
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 viewerA033298 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,995
Level class, k > L32,091 · 32.09 %
Weight class, k ≤ L67,904 · 67.91 %
Ties, k = L0
On the level line L = 12
Forced level, l ≤ d²33
Range of a(n)1 … 5,477,613
Range of the jump d1 … 99
Largest weight k, level L1,825,781, 235,788

13 different gaps occur, from 1 to 99; the level share is 32.09 %; L = 3 holds 32 % of the level class; 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.