decompwlj 3D

Numbers containing only digits coprime to 10 in their decimal representation

A136333 on the OEIS · family digit rule

Weight–level plate of Numbers containing only digits coprime to 10 in their decimal representation
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 viewerA136333 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,984
Level class, k > L17,520 · 17.52 %
Weight class, k ≤ L82,464 · 82.48 %
Ties, k = L15
On the level line L = 111,333
Forced level, l ≤ d²546
Range of a(n)1 … 119,133,179
Range of the jump d2 … 31,111,112
Largest weight k, level L119,131,967, 39,711,045

16 different gaps occur, from 2 to 31,111,112; the level share is 17.52 %; L = 1 holds 65 % of the level class; 16 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.