decompwlj 3D

23-smooth numbers: numbers whose prime divisors are all <= 23

A080683 on the OEIS · family smooth

Weight–level plate of 23-smooth numbers: numbers whose prime divisors are all <= 23
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 viewerA080683 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,998
Level class, k > L41,495 · 41.50 %
Weight class, k ≤ L58,503 · 58.50 %
Ties, k = L13
On the level line L = 1435
Forced level, l ≤ d²7,415
Range of a(n)1 … 397,393,920
Range of the jump d1 … 69,696
Largest weight k, level L347,486,089, 9,545,217

11,847 different gaps occur, from 1 to 69,696; the level share is 41.50 %; 7.4 % 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.