decompwlj 3D

Numbers such that the arithmetic mean of the cubes of their prime factors (taken with multiplicity) is a prime

A134619 on the OEIS · family multiplicative

Weight–level plate of Numbers such that the arithmetic mean of the cubes of their prime factors (taken with multiplicity) is a prime
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 viewerA134619 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,997
Level class, k > L39,485 · 39.49 %
Weight class, k ≤ L60,512 · 60.51 %
Ties, k = L2
On the level line L = 19,156
Forced level, l ≤ d²144
Range of a(n)20 … 10,347,252
Range of the jump d1 … 1,188
Largest weight k, level L10,346,983, 5,169,621

768 different gaps occur, from 1 to 1,188; the level share is 39.49 %.

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.