decompwlj 3D

Prime-indexed primes: primes with prime subscripts

A006450 on the OEIS · family primes · also known as Prime-indexed primes

Weight–level plate of Prime-indexed primes: primes with prime subscripts
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 viewerA006450 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,996
Level class, k > L43,290 · 43.29 %
Weight class, k ≤ L56,706 · 56.71 %
Ties, k = L10
On the level line L = 18,340
Forced level, l ≤ d²177
Range of a(n)3 … 20,491,057
Range of the jump d2 … 1,852
Largest weight k, level L20,489,699, 2,849,885

prime(prime(n)). The gaps are larger and more varied than the primes' (662 distinct values here), and the level share rises to 43.29 %, against 23.00 % for the primes.

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.