decompwlj 3D

Numbers k such that k, k+1 and k+2 all have the same number of distinct prime divisors

A006073 on the OEIS · family multiplicative

Weight–level plate of Numbers k such that k, k+1 and k+2 all have the same number of distinct prime divisors
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 viewerA006073 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,996
Level class, k > L22,036 · 22.04 %
Weight class, k ≤ L77,960 · 77.96 %
Ties, k = L21
On the level line L = 18,077
Forced level, l ≤ d²18
Range of a(n)2 … 1,416,788
Range of the jump d1 … 187
Largest weight k, level L1,416,577, 708,331

155 different gaps occur, from 1 to 187; the level share is 22.04 %; L = 1 holds 37 % 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.