decompwlj 3D

Sum of the first n primes

A007504 on the OEIS · family summatory

Weight–level plate of Sum of the first n primes
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 viewerA007504 on the OEIS
Terms100,000 (n = 0 … 99,999)
Decomposable (a > 2d)99,995
Level class, k > L99,995 · 100.00 %
Weight class, k ≤ L0 · 0.00 %
Ties, k = L0
On the level line L = 15,651
Forced level, l ≤ d²99,995
Range of a(n)0 … 62,259,399,012
Range of the jump d2 … 1,299,709
Largest weight k, level L62,226,911,531, 47,792

Sum of the first n primes: a ~ n^2 log n / 2 and d = p(n+1), so l/d^2 -> 0 like 1/(2 log n): every decomposable term is forced level (100 %).

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.