decompwlj 3D

Sub-perfect powers: perfect powers (squares, cubes etc., not including 1) minus 1

A045542 on the OEIS · family powers

Weight–level plate of Sub-perfect powers: perfect powers (squares, cubes etc., not including 1) minus 1
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 viewerA045542 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,997
Level class, k > L99,306 · 99.31 %
Weight class, k ≤ L691 · 0.69 %
Ties, k = L1
On the level line L = 17,425
Forced level, l ≤ d²97,799
Range of a(n)3 … 9,565,231,203
Range of the jump d1 … 195,605
Largest weight k, level L9,564,448,801, 6,981,034

97,727 different gaps occur, from 1 to 195,605; the level share is 99.31 %; 97.8 % of terms are forced level (l <= d^2).

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.