decompwlj 3D

Sum of totient function: a(n) = Sum_{k=1..n} phi(k), cf. A000010

A002088 on the OEIS · family summatory · also known as Sum of totients

Weight–level plate of Sum of totient function: a(n) = Sum_{k=1..n} phi(k), cf. A000010
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 viewerA002088 on the OEIS
Terms100,000 (n = 0 … 99,999)
Decomposable (a > 2d)99,994
Level class, k > L87,342 · 87.35 %
Weight class, k ≤ L12,652 · 12.65 %
Ties, k = L4
On the level line L = 10
Forced level, l ≤ d²45,973
Range of a(n)0 … 3,039,610,754
Range of the jump d1 … 99,990
Largest weight k, level L1,519,415,519, 133,258

a(n) = sum of phi(k) for k <= n, from a(0) = 0; a ~ 3n^2/pi^2 and d = phi(n+1) <= n, so the sequence is nearly quadratic. l <= d^2 on 45.98 % of terms and 87.35 % are level-classified. For n >= 2 both a and d are even, so 2 | l. Since a > 3d on every decomposable term, l/2 > d, so k <= l/2 and L >= 2: the line L = 1 is empty.

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.