decompwlj 3D

Primes that can be written as the sum of 13 consecutive primes

A127341 on the OEIS · family primes

Weight–level plate of Primes that can be written as the sum of 13 consecutive primes
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 viewerA127341 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)100,000
Level class, k > L57,633 · 57.63 %
Weight class, k ≤ L42,367 · 42.37 %
Ties, k = L2
On the level line L = 18,007
Forced level, l ≤ d²3,292
Range of a(n)691 … 176,579,167
Range of the jump d68 … 20,194
Largest weight k, level L176,554,219, 1,909,285

4,608 different gaps occur, from 68 to 20,194; the level share is 57.63 %; 3.3 % 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.