decompwlj 3D

3-almost primes indexed by primes

A124269 on the OEIS · family multiplicative

Weight–level plate of 3-almost primes indexed by primes
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 viewerA124269 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,997
Level class, k > L33,637 · 33.64 %
Weight class, k ≤ L66,360 · 66.36 %
Ties, k = L12
On the level line L = 18,334
Forced level, l ≤ d²42
Range of a(n)12 … 5,274,422
Range of the jump d2 … 450
Largest weight k, level L5,274,359, 1,746,241

372 different gaps occur, from 2 to 450; the level share is 33.64 %.

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.