decompwlj 3D

Primes of form x^2+53*y^2

A033234 on the OEIS · family quadratic form

Weight–level plate of Primes of form x^2+53*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 viewerA033234 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,997
Level class, k > L41,472 · 41.47 %
Weight class, k ≤ L58,525 · 58.53 %
Ties, k = L14
On the level line L = 17,908
Forced level, l ≤ d²286
Range of a(n)53 … 18,850,933
Range of the jump d4 … 1,992
Largest weight k, level L18,844,549, 3,748,701

369 different gaps occur, from 4 to 1,992; the level share is 41.47 %.

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.