decompwlj 3D

Prime divisors of Mersenne numbers. Primes p such that the multiplicative order of 2 modulo p is prime

A122094 on the OEIS · family primes

Weight–level plate of Prime divisors of Mersenne numbers. Primes p such that the multiplicative order of 2 modulo p is prime
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 viewerA122094 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,996
Level class, k > L48,930 · 48.93 %
Weight class, k ≤ L51,066 · 51.07 %
Ties, k = L0
On the level line L = 110,643
Forced level, l ≤ d²621
Range of a(n)3 … 46,082,447
Range of the jump d2 … 5,610
Largest weight k, level L46,081,127, 15,278,055

1,051 different gaps occur, from 2 to 5,610; the level share is 48.93 %; 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.