decompwlj 3D

Ulam numbers: a(1) = 1; a(2) = 2; for n>2, a(n) = least number > a(n-1) which is a unique sum of two distinct earlier terms

A002858 on the OEIS · family self-referential · also known as Ulam numbers

Weight–level plate of Ulam numbers: a(1) = 1; a(2) = 2; for n>2, a(n) = least number > a(n-1) which is a unique sum of two distinct earlier terms
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 viewerA002858 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,997
Level class, k > L21,855 · 21.86 %
Weight class, k ≤ L78,142 · 78.14 %
Ties, k = L19
On the level line L = 17,615
Forced level, l ≤ d²24
Range of a(n)1 … 1,351,223
Range of the jump d1 … 587
Largest weight k, level L1,351,199, 450,407

Rigid: gap 2 alone carries 37 % of the terms. The signal at alpha = 2.5714475 acts on the classification only through the gap - across phase bins mean gap and level share correlate at r = 0.9957.

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.