decompwlj 3D

Biquadrateful numbers

A046101 on the OEIS · family multiplicative

Weight–level plate of Biquadrateful numbers
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 viewerA046101 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,998
Level class, k > L13,701 · 13.70 %
Weight class, k ≤ L86,297 · 86.30 %
Ties, k = L6
On the level line L = 11,387
Forced level, l ≤ d²14
Range of a(n)16 … 1,314,720
Range of the jump d1 … 16
Largest weight k, level L1,314,539, 657,031

16 different gaps occur, from 1 to 16; the level share is 13.70 %; L = 16 holds 46 % of the level class.

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.