decompwlj 3D

Numbers with exactly 5 ones in binary expansion

A014313 on the OEIS · family binary rule

Weight–level plate of Numbers with exactly 5 ones in binary expansion
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 viewerA014313 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,999
Level class, k > L8,888 · 8.89 %
Weight class, k ≤ L91,111 · 91.11 %
Ties, k = L5
On the level line L = 12,365
Forced level, l ≤ d²3,303
Range of a(n)31 … 268,480,516
Range of the jump d1 … 8,388,623
Largest weight k, level L268,476,431, 134,240,256

106 different gaps occur, from 1 to 8,388,623; the level share is 8.89 %; 3.3 % 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.