decompwlj 3D

Primitive abundant numbers (abundant numbers all of whose proper divisors are deficient numbers)

A071395 on the OEIS · family divisor functions

Weight–level plate of Primitive abundant numbers (abundant numbers all of whose proper divisors are deficient 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 viewerA071395 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,998
Level class, k > L53,793 · 53.79 %
Weight class, k ≤ L46,205 · 46.21 %
Ties, k = L2
On the level line L = 12,296
Forced level, l ≤ d²6,886
Range of a(n)20 … 338,899,712
Range of the jump d1 … 59,981
Largest weight k, level L338,842,697, 127,251,247

11,641 different gaps occur, from 1 to 59,981; the level share is 53.79 %; 6.9 % of terms are forced level (l <= d^2).

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.