decompwlj 3D

Primes sandwiched between two numbers having same number of divisors

A067889 on the OEIS · family primes

Weight–level plate of Primes sandwiched between two numbers having same number of divisors
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 viewerA067889 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,995
Level class, k > L44,091 · 44.09 %
Weight class, k ≤ L55,904 · 55.91 %
Ties, k = L3
On the level line L = 17,740
Forced level, l ≤ d²472
Range of a(n)7 … 29,753,261
Range of the jump d2 … 3,486
Largest weight k, level L29,749,241, 9,904,935

1,011 different gaps occur, from 2 to 3,486; the level share is 44.09 %.

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.