decompwlj 3D

13-almost primes (generalization of semiprimes)

A069274 on the OEIS · family multiplicative

Weight–level plate of 13-almost primes (generalization of semiprimes)
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 viewerA069274 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,998
Level class, k > L24,623 · 24.62 %
Weight class, k ≤ L75,375 · 75.38 %
Ties, k = L5
On the level line L = 127
Forced level, l ≤ d²2,867
Range of a(n)8,192 … 151,853,056
Range of the jump d2 … 15,408
Largest weight k, level L147,913,807, 28,156,312

2,429 different gaps occur, from 2 to 15,408; the level share is 24.62 %; 2.9 % 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.