decompwlj 3D

Primes of the form 6k-1

A007528 on the OEIS · family primes · also known as Primes congruent to 5 mod 6

Weight–level plate of Primes of the form 6k-1
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 viewerA007528 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,998
Level class, k > L35,164 · 35.16 %
Weight class, k ≤ L64,834 · 64.84 %
Ties, k = L0
On the level line L = 117,780
Forced level, l ≤ d²20
Range of a(n)5 … 2,748,191
Range of the jump d6 … 258
Largest weight k, level L2,748,131, 392,261

Every term is 5 mod 6 and every gap 0 mod 6, so l = 5 (mod 6). No square is 5 mod 6: zero ties, by proof. The level share is 35.16 %, and half the level class sits on L = 1.

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.