decompwlj 3D

Numbers with digits in nondecreasing order

A009994 on the OEIS · family digit rule · also known as Nondecreasing digits

Weight–level plate of Numbers with digits in nondecreasing order
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 viewerA009994 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,997
Level class, k > L19,844 · 19.84 %
Weight class, k ≤ L80,153 · 80.16 %
Ties, k = L6
On the level line L = 18,580
Forced level, l ≤ d²2,655
Range of a(n)0 … 11,122,357,788
Range of the jump d1 … 1,111,111,112
Largest weight k, level L11,122,356,799, 5,561,178,888

Decimal digits never decrease (0, 1, ..., 9, 11, 12, ...). There are only C(d+9, 9) of them with d digits or fewer, so the terms grow fast: the 100,000th has 11 digits. The level share is 19.84 %; 2.7 % of terms are forced level (l <= d^2); L = 1 holds 43 % of the level class.

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.