decompwlj 3D

Numbers whose product of digits is a power of 2

A028846 on the OEIS · family digit rule

Weight–level plate of Numbers whose product of digits is a power of 2
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 viewerA028846 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,990
Level class, k > L5,156 · 5.16 %
Weight class, k ≤ L94,834 · 94.84 %
Ties, k = L0
On the level line L = 121
Forced level, l ≤ d²465
Range of a(n)1 … 118,122,148
Range of the jump d1 … 32,222,223
Largest weight k, level L66,666,665, 59,061,070

25 different gaps occur, from 1 to 32,222,223; the level share is 5.16 %; L = 5 holds 62 % of the level class; there are no ties; 10 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.