decompwlj 3D

Happy numbers: numbers whose trajectory under iteration of sum of squares of digits map (see A003132) includes 1

A007770 on the OEIS · family digit rule · also known as Happy numbers

Weight–level plate of Happy numbers: numbers whose trajectory under iteration of sum of squares of digits map (see A003132) includes 1
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 viewerA007770 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,999
Level class, k > L19,709 · 19.71 %
Weight class, k ≤ L80,290 · 80.29 %
Ties, k = L20
On the level line L = 18,508
Forced level, l ≤ d²16
Range of a(n)1 … 692,961
Range of the jump d1 … 73
Largest weight k, level L692,957, 346,447

The digit-square-sum iteration reaches 1. The gaps are irregular (58 distinct values up to 73) and the level share is 19.71 %.

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.