decompwlj 3D

Odd perfect powers (1 together with numbers m^k, m odd, k >= 2)

A075109 on the OEIS · family powers

Weight–level plate of Odd perfect powers (1 together with numbers m^k, m odd, 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 viewerA075109 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,996
Level class, k > L99,734 · 99.74 %
Weight class, k ≤ L262 · 0.26 %
Ties, k = L0
On the level line L = 120,430
Forced level, l ≤ d²99,147
Range of a(n)1 … 38,621,289,529
Range of the jump d2 … 786,096
Largest weight k, level L38,617,359,161, 61,471,557

99,140 different gaps occur, from 2 to 786,096; the level share is 99.74 %; 99.2 % 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.