decompwlj 3D

a(n) = n + Sum_{k=1..n} pi(k), where pi() = A000720

A002815 on the OEIS · family summatory

Weight–level plate of a(n) = n + Sum_{k=1..n} pi(k), where pi() = A000720
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 viewerA002815 on the OEIS
Terms100,000 (n = 0 … 99,999)
Decomposable (a > 2d)99,996
Level class, k > L70,237 · 70.24 %
Weight class, k ≤ L29,759 · 29.76 %
Ties, k = L11
On the level line L = 18,418
Forced level, l ≤ d²8
Range of a(n)0 … 504,903,462
Range of the jump d1 … 9,593
Largest weight k, level L504,778,759, 52,544

9,593 different gaps occur, from 1 to 9,593; the level share is 70.24 %.

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.