decompwlj 3D

Numbers such that the sum of squares of their prime factors (taken with multiplicity) is a prime

A134616 on the OEIS · family multiplicative

Weight–level plate of Numbers such that the sum of squares of their prime factors (taken with multiplicity) is a prime
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 viewerA134616 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,998
Level class, k > L24,746 · 24.75 %
Weight class, k ≤ L75,252 · 75.25 %
Ties, k = L6
On the level line L = 14,033
Forced level, l ≤ d²19
Range of a(n)6 … 1,988,710
Range of the jump d1 … 240
Largest weight k, level L1,987,429, 993,154

174 different gaps occur, from 1 to 240; the level share is 24.75 %.

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.