decompwlj 3D

Values taken by totient function phi(m) (A000010)

A002202 on the OEIS · family divisor functions · also known as Totients (values of phi)

Weight–level plate of Values taken by totient function phi(m) (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 viewerA002202 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,997
Level class, k > L12,584 · 12.58 %
Weight class, k ≤ L87,413 · 87.42 %
Ties, k = L1
On the level line L = 12
Forced level, l ≤ d²2
Range of a(n)1 … 539,388
Range of the jump d1 … 32
Largest weight k, level L269,597, 179,766

Values of Euler's phi. Past 1 every totient is even, so 2 | l and the line L = 1 holds 2 terms. The level share is 12.58 %; L = 4 holds 40 % of the level class.

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.