Smith (or joke) numbers: composite numbers k such that sum of digits of k = sum of digits of prime factors of k (counted with multiplicity)

Weight–level plate of Smith (or joke) numbers: composite numbers k such that sum of digits of k = sum of digits of prime factors of k (counted with multiplicity) — 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 A006753 on the OEIS

Terms	100,000 (n = 1 … 100,000)
Decomposable (a > 2d)	99,998
Level class, k > L	29,155 · 29.16 %
Weight class, k ≤ L	70,843 · 70.84 %
Ties, k = L	7
On the level line L = 1	6,493
Forced level, l ≤ d²	69
Range of a(n)	4 … 3,702,471
Range of the jump d	1 … 879
Largest weight k, level L	3,701,479, 1,850,842

429 different gaps occur, from 1 to 879; the level share is 29.16 %.

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.