decompwlj 3D

Lucky numbers

A000959 on the OEIS · family sieve

Weight–level plate of Lucky numbers
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 viewerA000959 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,998
Level class, k > L32,314 · 32.31 %
Weight class, k ≤ L67,684 · 67.69 %
Ties, k = L25
On the level line L = 119,084
Forced level, l ≤ d²8
Range of a(n)1 … 1,429,431
Range of the jump d2 … 144
Largest weight k, level L1,429,403, 285,865

A sieve of the same shape as Eratosthenes, with the same density x / log x. As for the primes, every term is odd and every gap even, so l is odd and every weight is odd. The level share is 32.31 %, against 23.00 % for the primes over a comparable range (a up to 1.43e6 and 1.30e6).

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.