decompwlj 3D

Numbers whose digits are squares

A046030 on the OEIS · family digit rule

Weight–level plate of Numbers whose digits are squares
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 viewerA046030 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,983
Level class, k > L15,560 · 15.56 %
Weight class, k ≤ L84,423 · 84.44 %
Ties, k = L9
On the level line L = 17,109
Forced level, l ≤ d²438
Range of a(n)0 … 140,144,199
Range of the jump d1 … 40,000,001
Largest weight k, level L140,144,189, 70,072,054

17 different gaps occur, from 1 to 40,000,001; the level share is 15.56 %; L = 1 holds 46 % of the level class; 17 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.