decompwlj 3D

Primes p such that q-p = 40, where q is the next prime after p

A126721 on the OEIS · family primes

Weight–level plate of Primes p such that q-p = 40, where q is the next prime after p
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 viewerA126721 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)100,000
Level class, k > L60,421 · 60.42 %
Weight class, k ≤ L39,579 · 39.58 %
Ties, k = L9
On the level line L = 114,761
Forced level, l ≤ d²3,149
Range of a(n)19,333 … 161,322,547
Range of the jump d42 … 19,320
Largest weight k, level L161,312,257, 3,577,069

1,696 different gaps occur, from 42 to 19,320; the level share is 60.42 %; 3.1 % 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.