decompwlj 3D

Partial sums of sequence A001221 (number of distinct primes dividing n)

A013939 on the OEIS · family summatory

Weight–level plate of Partial sums of sequence A001221 (number of distinct primes dividing n)
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 viewerA013939 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,996
Level class, k > L15,331 · 15.33 %
Weight class, k ≤ L84,665 · 84.67 %
Ties, k = L31
On the level line L = 18,757
Forced level, l ≤ d²0
Range of a(n)0 … 266,400
Range of the jump d1 … 6
Largest weight k, level L266,381, 133,187

6 different gaps occur, from 1 to 6; the level share is 15.33 %; L = 1 holds 57 % of the level class.

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.