decompwlj 3D

a(n) = -1 + Sum_{i=1..n} phi(i)

A015614 on the OEIS · family summatory

Weight–level plate of a(n) = -1 + Sum_{i=1..n} phi(i)
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 viewerA015614 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,995
Level class, k > L93,273 · 93.28 %
Weight class, k ≤ L6,722 · 6.72 %
Ties, k = L8
On the level line L = 110,496
Forced level, l ≤ d²45,974
Range of a(n)0 … 3,039,650,753
Range of the jump d1 … 99,990
Largest weight k, level L3,039,445,957, 131,525

18,412 different gaps occur, from 1 to 99,990; the level share is 93.28 %; 46.0 % of terms are forced level (l <= d^2).

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.