decompwlj 3D

Numbers of the form m * 2^bigomega(m), where m>1 is odd and bigomega(m) = A001222(m), the number of prime factors of m

A072978 on the OEIS · family multiplicative

Weight–level plate of Numbers of the form m * 2^bigomega(m), where m>1 is odd and bigomega(m) = A001222(m), the number of prime factors of m
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 viewerA072978 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,997
Level class, k > L14,040 · 14.04 %
Weight class, k ≤ L85,957 · 85.96 %
Ties, k = L0
On the level line L = 11
Forced level, l ≤ d²11
Range of a(n)1 … 934,658
Range of the jump d2 … 88
Largest weight k, level L467,003, 311,528

41 different gaps occur, from 2 to 88; the level share is 14.04 %; L = 2 holds 34 % of the level class; there are 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.