decompwlj 3D

Skip 1, take 1, skip 2, take 2, skip 3, take 3, etc

A004202 on the OEIS · family block · also known as Skip 1, take 1, skip 2, take 2, ...

Weight–level plate of Skip 1, take 1, skip 2, take 2, skip 3, take 3, etc
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 viewerA004202 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,998
Level class, k > L9,330 · 9.33 %
Weight class, k ≤ L90,668 · 90.67 %
Ties, k = L86
On the level line L = 18,971
Forced level, l ≤ d²444
Range of a(n)2 … 200,128
Range of the jump d1 … 448
Largest weight k, level L200,117, 100,063

Closed form { m^2+j : 1 <= j <= m }. Interior terms (d = 1) follow the naturals rule; block openers m^2+1 give a tie exactly when m is prime; every block closer satisfies l <= d^2 and is 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.