decompwlj 3D

Numbers that can be represented as a string of three or more 1's in a base >= 2

A053696 on the OEIS · family powers

Weight–level plate of Numbers that can be represented as a string of three or more 1's in a base >= 2
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 viewerA053696 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,999
Level class, k > L99,397 · 99.40 %
Weight class, k ≤ L602 · 0.60 %
Ties, k = L1
On the level line L = 119,006
Forced level, l ≤ d²97,323
Range of a(n)7 … 9,473,810,223
Range of the jump d1 … 194,668
Largest weight k, level L9,472,252,949, 305,175,781

96,955 different gaps occur, from 1 to 194,668; the level share is 99.40 %; 97.3 % 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.