decompwlj 3D

Lesser of twin primes

A001359 on the OEIS · family primes

Weight–level plate of Lesser of twin 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 viewerA001359 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,996
Level class, k > L47,766 · 47.77 %
Weight class, k ≤ L52,230 · 52.23 %
Ties, k = L0
On the level line L = 116,811
Forced level, l ≤ d²183
Range of a(n)3 … 18,409,199
Range of the jump d2 … 2,190
Largest weight k, level L18,408,323, 1,665,595

Past 5 every term is 5 mod 6, so d = 0 and l = 5 (mod 6). No square is 5 mod 6, so k = L can never happen: zero ties, by proof rather than by count. l is prime to 6, so every weight is odd and not a multiple of 3. The level share is 47.77 %.

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.