decompwlj 3D

Numbers having only digits 1 and 3 in their decimal representation

A032917 on the OEIS · family digit rule

Weight–level plate of Numbers having only digits 1 and 3 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 viewerA032917 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,969
Level class, k > L13,999 · 14.00 %
Weight class, k ≤ L85,970 · 86.00 %
Ties, k = L1
On the level line L = 14,027
Forced level, l ≤ d²804
Range of a(n)1 … 3,111,133,131,311,113
Range of the jump d2 … 1,777,777,777,777,778
Largest weight k, level L3,111,133,131,131,329, 1,037,044,377,103,703

31 different gaps occur, from 2 to 1,777,777,777,777,778; the level share is 14.00 %; L = 5 holds 30 % of the level class; 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.