decompwlj 3D

The odd numbers: a(n) = 2*n + 1

A005408 on the OEIS · family forced divisor · also known as Odd numbers

Weight–level plate of The odd numbers: a(n) = 2*n + 1
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 viewerA005408 on the OEIS
Terms100,000 (n = 0 … 99,999)
Decomposable (a > 2d)99,998
Level class, k > L17,982 · 17.98 %
Weight class, k ≤ L82,016 · 82.02 %
Ties, k = L85
On the level line L = 117,982
Forced level, l ≤ d²1
Range of a(n)1 … 199,999
Range of the jump d2 … 2
Largest weight k, level L199,967, 66,665

d = 2 and l = a - 2 is odd, so the divisor 2 is never available and k = spf(a - 2). The level class is exactly {a : a - 2 prime}, all on L = 1: 17,982 = pi(199,997) - 1 terms here. The ties are a - 2 = p^2, 85 = pi(447) - 1. Removing the divisor 2 nearly doubles the level share against the naturals: 17.98 % against 9.59 %.

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.