decompwlj 3D

Pythagorean primes: primes of the form 4*k + 1

A002144 on the OEIS · family primes · also known as Primes congruent to 1 mod 4

Weight–level plate of Pythagorean primes: primes of the form 4*k + 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 viewerA002144 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,998
Level class, k > L27,951 · 27.95 %
Weight class, k ≤ L72,047 · 72.05 %
Ties, k = L23
On the level line L = 17,055
Forced level, l ≤ d²21
Range of a(n)5 … 2,752,517
Range of the jump d4 … 248
Largest weight k, level L2,752,153, 550,221

Every term is 1 mod 4 and every gap 0 mod 4, so l = 1 (mod 4) and ties are possible (23 here), where the primes 3 mod 4 have none. The level share, 27.95 %, is close to theirs (28.16 %). The fullest level line is L = 3.

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.