decompwlj 3D

Primes of the form 8k + 5

A007521 on the OEIS · family primes · also known as Primes congruent to 5 mod 8

Weight–level plate of Primes of the form 8k + 5
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 viewerA007521 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,997
Level class, k > L33,290 · 33.29 %
Weight class, k ≤ L66,707 · 66.71 %
Ties, k = L0
On the level line L = 16,878
Forced level, l ≤ d²62
Range of a(n)5 … 5,799,149
Range of the jump d8 … 616
Largest weight k, level L5,796,061, 644,005

Primes 5 mod 8. Every gap is 0 mod 8, so l = 5 (mod 8); no square is 5 mod 8, so there are no ties, by proof. The level share is 33.29 %.

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.