decompwlj 3D

Numbers m such that the factorizations of m..m+2 have the same number of primes (including multiplicities)

A045939 on the OEIS · family multiplicative

Weight–level plate of Numbers m such that the factorizations of m..m+2 have the same number of primes (including multiplicities)
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 viewerA045939 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,999
Level class, k > L30,388 · 30.39 %
Weight class, k ≤ L69,611 · 69.61 %
Ties, k = L11
On the level line L = 19,038
Forced level, l ≤ d²84
Range of a(n)33 … 4,323,181
Range of the jump d1 … 501
Largest weight k, level L4,323,047, 2,160,976

375 different gaps occur, from 1 to 501; the level share is 30.39 %.

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.