decompwlj 3D

Deficient numbers: numbers k such that sigma(k) < 2k

A005100 on the OEIS · family divisor functions · also known as Deficient numbers

Weight–level plate of Deficient numbers: numbers k such that sigma(k) < 2k
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 viewerA005100 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,998
Level class, k > L7,789 · 7.79 %
Weight class, k ≤ L92,209 · 92.21 %
Ties, k = L92
On the level line L = 17,789
Forced level, l ≤ d²1
Range of a(n)1 … 132,937
Range of the jump d1 … 3
Largest weight k, level L132,857, 66,468

sigma(n) < 2n. Gaps 1 and 2, with 38 gaps of 3. Every level-classified term lies on L = 1, and the level share, 7.79 %, is the lowest here after the numbers not divisible by 3.

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.