decompwlj 3D

Numbers congruent to 1 or 5 mod 6

A007310 on the OEIS · family forced divisor · also known as Numbers coprime to 6

Weight–level plate of Numbers congruent to 1 or 5 mod 6
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 viewerA007310 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,998
Level class, k > L9,593 · 9.59 %
Weight class, k ≤ L90,405 · 90.41 %
Ties, k = L2
On the level line L = 12
Forced level, l ≤ d²3
Range of a(n)1 … 299,999
Range of the jump d2 … 4
Largest weight k, level L99,991, 99,999

6m + 1 and 6m + 5, with gaps 4 and 2. Then 3 | l at every term, so after a gap of 2 the weight is 3. After a gap of 4, a term is level essentially when l/3 is prime. The level class is the naturals' level line moved to L = 3 (9.59 % level).

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.