decompwlj 3D

Numbers n such that phi(n) = phi(n+12), with Euler's totient function phi = A000010

A217139 on the OEIS · family divisor functions

Weight–level plate of Numbers n such that phi(n) = phi(n+12), with Euler's totient function phi = 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 viewerA217139 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)100,000
Level class, k > L38,174 · 38.17 %
Weight class, k ≤ L61,826 · 61.83 %
Ties, k = L2
On the level line L = 184
Forced level, l ≤ d²561
Range of a(n)48 … 45,281,832
Range of the jump d1 … 5,566
Largest weight k, level L44,211,073, 11,301,040

1,858 different gaps occur, from 1 to 5,566; the level share is 38.17 %.

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.