decompwlj 3D

Numbers n such that the decimal expansions of both n and n^2 have 1 as smallest digit and 7 as largest digit

A257210 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 7 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 viewerA257210 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,998
Level class, k > L34,420 · 34.42 %
Weight class, k ≤ L65,578 · 65.58 %
Ties, k = L0
On the level line L = 15,578
Forced level, l ≤ d²2,288
Range of a(n)271 … 1,522,344,761
Range of the jump d1 … 353,335,065
Largest weight k, level L1,522,323,151, 737,862,582

7,384 different gaps occur, from 1 to 353,335,065; the level share is 34.42 %; 2.3 % of terms are forced level (l <= d^2); there are no ties.

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.