decompwlj 3D

a(n+1) = next number having largest prime dividing a(n) as a factor, with a(1) = 2

A036441 on the OEIS · family self-referential

Weight–level plate of a(n+1) = next number having largest prime dividing a(n) as a factor, with a(1) = 2
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 viewerA036441 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,997
Level class, k > L99,997 · 100.00 %
Weight class, k ≤ L0 · 0.00 %
Ties, k = L0
On the level line L = 110,265
Forced level, l ≤ d²60,260
Range of a(n)2 … 2,500,099,997
Range of the jump d2 … 49,999
Largest weight k, level L2,499,900,001, 49,999

5,133 different gaps occur, from 2 to 49,999; the level share is 100.00 %; 60.3 % of terms are forced level (l <= d^2); there are no ties.

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.