decompwlj 3D

Numbers n such that the decimal expansions of both n and n^2 have 1 as smallest digit and 8 as largest digit

A257211 on the OEIS · family digit rule

Weight–level plate of Numbers n such that the decimal expansions of both n and n^2 have 1 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 viewerA257211 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,999
Level class, k > L24,202 · 24.20 %
Weight class, k ≤ L75,797 · 75.80 %
Ties, k = L4
On the level line L = 14,516
Forced level, l ≤ d²838
Range of a(n)128 … 28,522,818
Range of the jump d1 … 2,229,243
Largest weight k, level L28,521,523, 14,261,070

1,845 different gaps occur, from 1 to 2,229,243; the level share is 24.20 %.

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.