decompwlj 3D

Primes of the form x^2 + (x+3)^2

A076727 on the OEIS · family primes

Weight–level plate of Primes of the form x^2 + (x+3)^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 viewerA076727 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,993
Level class, k > L99,993 · 100.00 %
Weight class, k ≤ L0 · 0.00 %
Ties, k = L0
On the level line L = 116,202
Forced level, l ≤ d²99,993
Range of a(n)17 … 4,307,896,180,649
Range of the jump d12 … 975,329,472
Largest weight k, level L4,307,637,879,089, 716,645

98,702 different gaps occur, from 12 to 975,329,472; every decomposable term is forced level (l <= d^2); 7 terms do not decompose.

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.