decompwlj 3D

Numbers n such that every digit contains a loop (version 2)

A001744 on the OEIS · family digit rule

Weight–level plate of Numbers n such that every digit contains a loop (version 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 viewerA001744 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,991
Level class, k > L13,733 · 13.73 %
Weight class, k ≤ L86,258 · 86.27 %
Ties, k = L1
On the level line L = 13,177
Forced level, l ≤ d²304
Range of a(n)0 … 44,499,999
Range of the jump d1 … 30,000,001
Largest weight k, level L44,499,947, 22,249,994

16 different gaps occur, from 1 to 30,000,001; the level share is 13.73 %; L = 2 holds 48 % of the level class; 9 terms do not decompose.

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.