decompwlj 3D

Numbers using only digits 1 and 6

A284293 on the OEIS · family digit rule

Weight–level plate of Numbers using only digits 1 and 6
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 viewerA284293 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,969
Level class, k > L16,465 · 16.47 %
Weight class, k ≤ L83,504 · 83.53 %
Ties, k = L0
On the level line L = 13,971
Forced level, l ≤ d²1,122
Range of a(n)1 … 6,111,166,161,611,116
Range of the jump d5 … 4,444,444,444,444,445
Largest weight k, level L6,111,166,161,160,721, 1,018,527,693,601,851

16 different gaps occur, from 5 to 4,444,444,444,444,445; the level share is 16.47 %; 1.1 % of terms are forced level (l <= d^2); there are no ties; 31 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.