decompwlj 3D

Primes of the form 4*k + 3

A002145 on the OEIS · family primes · also known as Primes congruent to 3 mod 4

Weight–level plate of Primes of the form 4*k + 3
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 viewerA002145 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,997
Level class, k > L28,157 · 28.16 %
Weight class, k ≤ L71,840 · 71.84 %
Ties, k = L0
On the level line L = 17,048
Forced level, l ≤ d²20
Range of a(n)3 … 2,747,671
Range of the jump d4 … 256
Largest weight k, level L2,747,279, 549,471

Every term is 3 mod 4 and every gap 0 mod 4, so l = 3 (mod 4). No square is 3 mod 4: zero ties, by proof. l is odd, so every weight is odd. The level share, 28.16 %, is above the primes' 23.00 %.

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.