decompwlj 3D

Primes of the form x^2+101y^2

A139489 on the OEIS · family quadratic form

Weight–level plate of Primes of the form x^2+101y^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 viewerA139489 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,998
Level class, k > L46,892 · 46.89 %
Weight class, k ≤ L53,106 · 53.11 %
Ties, k = L6
On the level line L = 17,700
Forced level, l ≤ d²831
Range of a(n)101 … 46,575,961
Range of the jump d4 … 5,380
Largest weight k, level L46,574,257, 9,311,349

832 different gaps occur, from 4 to 5,380; the level share is 46.89 %.

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.