decompwlj 3D

a(n) = Sum_{k=1..n} floor(n/k); also Sum_{k=1..n} d(k), where d = number of divisors (A000005); also number of solutions to x*y = z with 1 <= x,y,z <= n

A006218 on the OEIS · family summatory · also known as Sum of d(k) for k <= n

Weight–level plate of a(n) = Sum_{k=1..n} floor(n/k); also Sum_{k=1..n} d(k), where d = number of divisors (A000005); also number of solutions to x*y = z with 1 <= x,y,z <= 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 viewerA006218 on the OEIS
Terms100,000 (n = 0 … 99,999)
Decomposable (a > 2d)99,996
Level class, k > L23,409 · 23.41 %
Weight class, k ≤ L76,587 · 76.59 %
Ties, k = L23
On the level line L = 17,854
Forced level, l ≤ d²3
Range of a(n)0 … 1,166,714
Range of the jump d1 … 128
Largest weight k, level L1,164,829, 388,858

Sum of d(k) for k <= n, ~ n log n; the gap is d(n+1) (1 to 128 here). The level share, 23.41 %, is close to the primes' 23.00 %.

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.