decompwlj 3D

Primes of the form 210n + 1

A073102 on the OEIS · family primes

Weight–level plate of Primes of the form 210n + 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 viewerA073102 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,997
Level class, k > L62,766 · 62.77 %
Weight class, k ≤ L37,231 · 37.23 %
Ties, k = L63
On the level line L = 124,332
Forced level, l ≤ d²1,249
Range of a(n)211 … 82,288,711
Range of the jump d210 … 7,350
Largest weight k, level L82,276,111, 384,091

34 different gaps occur, from 210 to 7,350; the level share is 62.77 %; 1.2 % of terms are forced level (l <= d^2); L = 1 holds 39 % of the level class.

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.