decompwlj 3D

Total number of 1's in binary expansions of 0, ..., n

A000788 on the OEIS · family summatory

Weight–level plate of Total number of 1's in binary expansions of 0, ..., n
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 viewerA000788 on the OEIS
Terms100,000 (n = 0 … 99,999)
Decomposable (a > 2d)99,997
Level class, k > L24,123 · 24.12 %
Weight class, k ≤ L75,874 · 75.88 %
Ties, k = L15
On the level line L = 18,227
Forced level, l ≤ d²2
Range of a(n)0 … 815,024
Range of the jump d1 … 16
Largest weight k, level L814,949, 195,242

16 different gaps occur, from 1 to 16; the level share is 24.12 %; L = 1 holds 34 % of the level class.

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.