decompwlj 3D

Numbers n such that BCR(n) = n, where BCR = binary-complement-and-reverse = take one's complement then reverse bit order

A035928 on the OEIS · family binary rule

Weight–level plate of Numbers n such that BCR(n) = n, where BCR = binary-complement-and-reverse = take one's complement then reverse bit order
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 viewerA035928 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,984
Level class, k > L84,417 · 84.43 %
Weight class, k ≤ L15,567 · 15.57 %
Ties, k = L2
On the level line L = 10
Forced level, l ≤ d²49,984
Range of a(n)2 … 13,107,328,316
Range of the jump d2 … 4,295,163,902
Largest weight k, level L6,441,545,729, 199,238

150 different gaps occur, from 2 to 4,295,163,902; the level share is 84.43 %; 50.0 % of terms are forced level (l <= d^2); 16 terms do not decompose.

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.