decompwlj 3D

Binary palindromes: numbers whose binary expansion is palindromic

A006995 on the OEIS · family binary rule · also known as Binary palindromes

Weight–level plate of Binary palindromes: numbers whose binary expansion is palindromic
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 viewerA006995 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,996
Level class, k > L88,397 · 88.40 %
Weight class, k ≤ L11,599 · 11.60 %
Ties, k = L3
On the level line L = 16,856
Forced level, l ≤ d²35,078
Range of a(n)0 … 2,258,634,081
Range of the jump d1 … 98,304
Largest weight k, level L2,147,205,119, 429,496,729

Binary palindromes; starts at a(1) = 0. There are about sqrt(x) of them up to x, so the sequence thins out like a quadratic. The gap outgrows sqrt(l) on 35 % of terms (forced level), and 88.40 % are level-classified.

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.