decompwlj 3D

Numbers with exactly 4 ones in binary expansion

A014312 on the OEIS · family binary rule

Weight–level plate of Numbers with exactly 4 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 viewerA014312 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,999
Level class, k > L12,941 · 12.94 %
Weight class, k ≤ L87,058 · 87.06 %
Ties, k = L68
On the level line L = 11,097
Forced level, l ≤ d²9,421
Range of a(n)15 … 1,374,390,059,012
Range of the jump d1 … 68,719,476,743
Largest weight k, level L1,305,670,057,981, 687,195,029,504

139 different gaps occur, from 1 to 68,719,476,743; the level share is 12.94 %; 9.4 % 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.