decompwlj 3D

Numbers with digits 3 and 4 only

A032834 on the OEIS · family digit rule

Weight–level plate of Numbers with digits 3 and 4 only
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 viewerA032834 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,985
Level class, k > L6,256 · 6.26 %
Weight class, k ≤ L93,729 · 93.74 %
Ties, k = L0
On the level line L = 116
Forced level, l ≤ d²628
Range of a(n)3 … 4,333,344,343,433,334
Range of the jump d1 … 2,888,888,888,888,889
Largest weight k, level L2,555,555,555,555,555, 2,166,672,171,716,666

31 different gaps occur, from 1 to 2,888,888,888,888,889; the level share is 6.26 %; L = 5 holds 33 % of the level class; there are no ties; 15 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.