decompwlj 3D

Numbers k such that k and k^2 have same digit sum

A058369 on the OEIS · family digit rule

Weight–level plate of Numbers k such that k and k^2 have same digit sum
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 viewerA058369 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,995
Level class, k > L33,841 · 33.84 %
Weight class, k ≤ L66,154 · 66.16 %
Ties, k = L7
On the level line L = 19,255
Forced level, l ≤ d²2,000
Range of a(n)0 … 28,672,992
Range of the jump d1 … 54,000
Largest weight k, level L28,670,591, 14,329,498

1,743 different gaps occur, from 1 to 54,000; the level share is 33.84 %; 2.0 % of terms are forced level (l <= d^2).

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.