decompwlj 3D

Numbers k such that floor(sqrt(k)) divides k

A006446 on the OEIS · family block · also known as Numbers k with floor(sqrt k) | k

Weight–level plate of Numbers k such that floor(sqrt(k)) divides k
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 viewerA006446 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,997
Level class, k > L70,046 · 70.05 %
Weight class, k ≤ L29,951 · 29.95 %
Ties, k = L0
On the level line L = 16,952
Forced level, l ≤ d²66,664
Range of a(n)1 … 1,111,155,556
Range of the jump d1 … 33,334
Largest weight k, level L1,110,955,561, 555,577,777

Blocks {m^2, m^2 + m, m^2 + 2m} with gaps m, m, 1. The first two terms of each block have l <= d^2 and are forced level - exactly two thirds of the sequence (66.67 %). The third, with d = 1 and l = (m+1)^2 - 2, never a square, behaves like a natural number: 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.