decompwlj 3D

Nontotients: even numbers k such that phi(m) = k has no solution

A005277 on the OEIS · family divisor functions

Weight–level plate of Nontotients: even numbers k such that phi(m) = k has no solution
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 viewerA005277 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,999
Level class, k > L10,729 · 10.73 %
Weight class, k ≤ L89,270 · 89.27 %
Ties, k = L0
On the level line L = 10
Forced level, l ≤ d²5
Range of a(n)14 … 322,794
Range of the jump d2 … 16
Largest weight k, level L161,387, 107,596

8 different gaps occur, from 2 to 16; the level share is 10.73 %; L = 2 holds 95 % of the level class; there are no ties.

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.