decompwlj 3D

Numbers of the form p^2 + q^2, with p and q primes

A045636 on the OEIS · family primes

Weight–level plate of Numbers of the form p^2 + q^2, with p and q primes
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 viewerA045636 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,998
Level class, k > L44,447 · 44.45 %
Weight class, k ≤ L55,551 · 55.55 %
Ties, k = L0
On the level line L = 1116
Forced level, l ≤ d²153
Range of a(n)8 … 16,783,658
Range of the jump d3 … 1,680
Largest weight k, level L16,752,647, 2,792,108

90 different gaps occur, from 3 to 1,680; the level share is 44.45 %; L = 2 holds 44 % of the level class; there are no ties.

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.