decompwlj 3D

Concatenation of two or more consecutive positive integers

A035333 on the OEIS · family digit rule

Weight–level plate of Concatenation of two or more consecutive positive integers
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 viewerA035333 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,995
Level class, k > L99,853 · 99.86 %
Weight class, k ≤ L142 · 0.14 %
Ties, k = L0
On the level line L = 15,459
Forced level, l ≤ d²99,628
Range of a(n)12 … 9,879,198,792
Range of the jump d10 … 2,222,121
Largest weight k, level L9,877,298,773, 3,106,144

106 different gaps occur, from 10 to 2,222,121; the level share is 99.86 %; 99.6 % of terms are forced level (l <= d^2); 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.