decompwlj 3D

Primes of form x^2+65*y^2

A033241 on the OEIS · family quadratic form

Weight–level plate of Primes of form x^2+65*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 viewerA033241 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,998
Level class, k > L47,851 · 47.85 %
Weight class, k ≤ L52,147 · 52.15 %
Ties, k = L20
On the level line L = 110,948
Forced level, l ≤ d²383
Range of a(n)101 … 25,651,001
Range of the jump d8 … 3,172
Largest weight k, level L25,650,881, 2,848,617

300 different gaps occur, from 8 to 3,172; the level share is 47.85 %.

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.