Economical numbers: write n as a product of primes raised to powers, let D(n) = number of digits in product, l(n) = number of digits in n; sequence gives n such that D(n) < l(n)

Weight–level plate of Economical numbers: write n as a product of primes raised to powers, let D(n) = number of digits in product, l(n) = number of digits in n; sequence gives n such that D(n) < l(n) — 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 viewer A046759 on the OEIS

Terms	100,000 (n = 1 … 100,000)
Decomposable (a > 2d)	99,999
Level class, k > L	41,216 · 41.22 %
Weight class, k ≤ L	58,783 · 58.78 %
Ties, k = L	11
On the level line L = 1	8,824
Forced level, l ≤ d²	744
Range of a(n)	125 … 23,037,440
Range of the jump d	1 … 8,438
Largest weight k, level L	23,034,679, 11,397,375

2,178 different gaps occur, from 1 to 8,438; the level share is 41.22 %.

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.