decompwlj 3D

Squares of primes

A001248 on the OEIS · family powers

Weight–level plate of Squares of 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 viewerA001248 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,994
Level class, k > L99,994 · 100.00 %
Weight class, k ≤ L0 · 0.00 %
Ties, k = L0
On the level line L = 117,945
Forced level, l ≤ d²99,994
Range of a(n)4 … 1,689,243,484,681
Range of the jump d5 … 238,596,672
Largest weight k, level L1,689,108,317,137, 315,511

a = p^2 and d = q^2 - p^2 >= 4p + 4, so l < p^2 < d^2: every decomposable term is forced level (100 %). a decomposes iff q^2 < 1.5 p^2. That fails only for p = 2, 3, 5, 7, 13, 23; Nagura (a prime in [x, 6x/5] for x >= 25) rules out any other.

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.