The positive integers. Also called the natural numbers, the whole numbers or the counting numbers, but these terms are ambiguous

A000027 on the OEIS · family base case · also known as Natural numbers

Weight–level plate of The positive integers. Also called the natural numbers, the whole numbers or the counting numbers, but these terms are ambiguous — 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 viewer A000027 on the OEIS

Terms	100,000 (n = 1 … 100,000)
Decomposable (a > 2d)	99,998
Level class, k > L	9,592 · 9.59 %
Weight class, k ≤ L	90,406 · 90.41 %
Ties, k = L	65
On the level line L = 1	9,592
Forced level, l ≤ d²	0
Range of a(n)	1 … 100,000
Range of the jump d	1 … 1
Largest weight k, level L	99,991, 49,999

The base case: d = 1 everywhere, so k = spf(a - 1) and the weight sheet is the sieve of Eratosthenes. The level class is the single line L = 1 (a - 1 prime): 9,592 = pi(99,999) terms here. The ties k = L are a - 1 = p^2: 65 = pi(316).

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.