decompwlj 3D

Primes congruent to {1, 3} (mod 8); or, odd primes of form x^2 + 2*y^2

A033200 on the OEIS · family primes

Weight–level plate of Primes congruent to {1, 3} (mod 8); or, odd primes of form x^2 + 2*y^2
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 viewerA033200 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,997
Level class, k > L28,462 · 28.46 %
Weight class, k ≤ L71,535 · 71.54 %
Ties, k = L7
On the level line L = 17,542
Forced level, l ≤ d²29
Range of a(n)3 … 2,752,843
Range of the jump d2 … 226
Largest weight k, level L2,752,637, 917,573

83 different gaps occur, from 2 to 226; the level share is 28.46 %.

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.