decompwlj 3D

Perfect powers: m^k where m > 0 and k >= 2

A001597 on the OEIS · family powers · also known as Perfect powers

Weight–level plate of Perfect powers: m^k where m > 0 and k >= 2
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 viewerA001597 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,996
Level class, k > L99,250 · 99.25 %
Weight class, k ≤ L746 · 0.75 %
Ties, k = L0
On the level line L = 19,879
Forced level, l ≤ d²97,797
Range of a(n)1 … 9,565,035,601
Range of the jump d1 … 195,603
Largest weight k, level L9,562,688,519, 28,812,787

1 and m^e with e >= 2. About sqrt(x) of them lie below x, so the sequence thins out like the squares, and 97.8 % of terms are forced level. The level share is 99.25 %; 97.8 % 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.