decompwlj 3D

Numbers n such that the decimal expansions of both n and n^2 have 2 as smallest digit and 8 as largest digit

A257368 on the OEIS · family digit rule

Weight–level plate of Numbers n such that the decimal expansions of both n and n^2 have 2 as smallest digit and 8 as largest digit
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 viewerA257368 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,986
Level class, k > L31,233 · 31.24 %
Weight class, k ≤ L68,753 · 68.76 %
Ties, k = L0
On the level line L = 12,299
Forced level, l ≤ d²1,671
Range of a(n)278 … 2,876,436,665
Range of the jump d1 … 1,397,334,049
Largest weight k, level L2,876,432,183, 1,438,186,737

6,808 different gaps occur, from 1 to 1,397,334,049; the level share is 31.24 %; 1.7 % of terms are forced level (l <= d^2); there are no ties; 14 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.