decompwlj 3D

Primes of the form k^2 + 3

A049423 on the OEIS · family primes

Weight–level plate of Primes of the form k^2 + 3
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 viewerA049423 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,991
Level class, k > L99,991 · 100.00 %
Weight class, k ≤ L0 · 0.00 %
Ties, k = L0
On the level line L = 116,160
Forced level, l ≤ d²99,991
Range of a(n)3 … 5,890,552,286,119
Range of the jump d4 … 1,016,645,616
Largest weight k, level L5,889,950,386,399, 576,559

98,756 different gaps occur, from 4 to 1,016,645,616; every decomposable term is forced level (l <= d^2); 9 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.