decompwlj 3D

13-smooth numbers: numbers whose prime divisors are all <= 13

A080197 on the OEIS · family smooth

Weight–level plate of 13-smooth numbers: numbers whose prime divisors are all <= 13
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 viewerA080197 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,998
Level class, k > L93,080 · 93.08 %
Weight class, k ≤ L6,918 · 6.92 %
Ties, k = L9
On the level line L = 1103
Forced level, l ≤ d²87,341
Range of a(n)1 … 476,636,015,625
Range of the jump d1 … 115,531,416
Largest weight k, level L413,572,961,953, 403,794

62,590 different gaps occur, from 1 to 115,531,416; the level share is 93.08 %; 87.3 % 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.