decompwlj 3D

Primes with indices that are primes with prime indices

A038580 on the OEIS · family primes

Weight–level plate of Primes with indices that are primes with prime indices
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 viewerA038580 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,995
Level class, k > L64,560 · 64.56 %
Weight class, k ≤ L35,435 · 35.44 %
Ties, k = L5
On the level line L = 17,286
Forced level, l ≤ d²6,530
Range of a(n)5 … 383,270,263
Range of the jump d6 … 36,468
Largest weight k, level L383,253,137, 4,062,269

7,843 different gaps occur, from 6 to 36,468; the level share is 64.56 %; 6.5 % 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.