decompwlj 3D

Primes of the form 512*k+1

A076339 on the OEIS · family primes

Weight–level plate of Primes of the form 512*k+1
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 viewerA076339 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)100,000
Level class, k > L64,205 · 64.20 %
Weight class, k ≤ L35,795 · 35.80 %
Ties, k = L8
On the level line L = 15,501
Forced level, l ≤ d²8,455
Range of a(n)7,681 … 485,306,881
Range of the jump d512 … 49,152
Largest weight k, level L485,277,697, 927,745

80 different gaps occur, from 512 to 49,152; the level share is 64.20 %; 8.5 % of terms are forced level (l <= d^2).

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.