decompwlj 3D

Primes of the form 2m*691 - 1

A134671 on the OEIS · family primes

Weight–level plate of Primes of the form 2m*691 - 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 viewerA134671 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,996
Level class, k > L73,150 · 73.15 %
Weight class, k ≤ L26,846 · 26.85 %
Ties, k = L0
On the level line L = 15,459
Forced level, l ≤ d²20,762
Range of a(n)1,381 … 1,378,699,783
Range of the jump d1,382 … 120,234
Largest weight k, level L1,378,094,467, 985,365

78 different gaps occur, from 1,382 to 120,234; the level share is 73.15 %; 20.8 % of terms are forced level (l <= d^2); there are no ties.

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.