decompwlj 3D

Semiprimes (or biprimes): products of two primes

A001358 on the OEIS · family multiplicative · also known as Semiprimes

Weight–level plate of Semiprimes (or biprimes): products of two 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 viewerA001358 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,998
Level class, k > L15,707 · 15.71 %
Weight class, k ≤ L84,291 · 84.29 %
Ties, k = L41
On the level line L = 19,563
Forced level, l ≤ d²5
Range of a(n)4 … 459,577
Range of the jump d1 … 47
Largest weight k, level L459,397, 229,770

Omega(n) = 2. The gaps stay small (1 to 47 here) and l/d^2 is large, so only 5 terms are forced level. The level share, 15.71 %, lies between the naturals' and the primes'. The line L = 1 (l prime) carries 61 % of the level class.

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.