decompwlj 3D

Numbers such that at least one digit contains a loop (version 2). Also called "holey" or "holy" numbers

A001745 on the OEIS · family digit rule

Weight–level plate of Numbers such that at least one digit contains a loop (version 2). Also called "holey" or "holy" numbers
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 viewerA001745 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,998
Level class, k > L9,763 · 9.76 %
Weight class, k ≤ L90,235 · 90.24 %
Ties, k = L51
On the level line L = 19,323
Forced level, l ≤ d²3
Range of a(n)0 … 103,904
Range of the jump d1 … 4
Largest weight k, level L103,903, 51,951

The gaps are 1, 2 and 4; the level share is 9.76 %; L = 1 holds 95 % 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.