decompwlj 3D

Evil primes: primes with even number of 1's in their binary expansion

A027699 on the OEIS · family primes

Weight–level plate of Evil primes: primes with even number of 1's in their binary expansion
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 viewerA027699 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,998
Level class, k > L30,294 · 30.29 %
Weight class, k ≤ L69,704 · 69.71 %
Ties, k = L6
On the level line L = 110,164
Forced level, l ≤ d²35
Range of a(n)3 … 2,868,793
Range of the jump d2 … 352
Largest weight k, level L2,868,787, 955,955

130 different gaps occur, from 2 to 352; the level share is 30.29 %; L = 1 holds 34 % 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.