decompwlj 3D

Numbers that are palindromic in base 8

A029803 on the OEIS · family digit rule

Weight–level plate of Numbers that are palindromic in base 8
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 viewerA029803 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,995
Level class, k > L84,848 · 84.85 %
Weight class, k ≤ L15,147 · 15.15 %
Ties, k = L19
On the level line L = 15,701
Forced level, l ≤ d²29,762
Range of a(n)0 … 2,203,075,778
Range of the jump d1 … 36,864
Largest weight k, level L2,147,385,337, 153,391,688

12 different gaps occur, from 1 to 36,864; the level share is 84.85 %; 29.8 % 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.