decompwlj 3D

Unhappy numbers: numbers having period-8 2-digitized sequences

A031177 on the OEIS · family digit rule · also known as Unhappy numbers

Weight–level plate of Unhappy numbers: numbers having period-8 2-digitized sequences
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 viewerA031177 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,999
Level class, k > L10,375 · 10.38 %
Weight class, k ≤ L89,624 · 89.62 %
Ties, k = L55
On the level line L = 19,539
Forced level, l ≤ d²1
Range of a(n)2 … 117,124
Range of the jump d1 … 6
Largest weight k, level L117,119, 58,561

The digit-square-sum iteration never reaches 1 (it falls into the cycle through 4). The level share is 10.38 %; L = 1 holds 92 % of the level class.

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.