decompwlj 3D

19-smooth numbers: numbers whose prime divisors are all <= 19

A080682 on the OEIS · family smooth

Weight–level plate of 19-smooth numbers: numbers whose prime divisors are all <= 19
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 viewerA080682 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,998
Level class, k > L55,594 · 55.60 %
Weight class, k ≤ L44,404 · 44.40 %
Ties, k = L9
On the level line L = 1291
Forced level, l ≤ d²29,926
Range of a(n)1 … 1,746,370,560
Range of the jump d1 … 333,151
Largest weight k, level L1,726,473,191, 11,569,960

22,333 different gaps occur, from 1 to 333,151; the level share is 55.60 %; 29.9 % 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.