decompwlj 3D

Let B(n) be the sum of binary digits of n. This sequence contains n such that B(n) = B(n^2)

A077436 on the OEIS · family binary rule

Weight–level plate of Let B(n) be the sum of binary digits of n. This sequence contains n such that B(n) = B(n^2)
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 viewerA077436 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,993
Level class, k > L28,465 · 28.47 %
Weight class, k ≤ L71,528 · 71.53 %
Ties, k = L3
On the level line L = 15,226
Forced level, l ≤ d²2,105
Range of a(n)0 … 24,692,471
Range of the jump d1 … 57,366
Largest weight k, level L24,689,947, 12,345,339

3,140 different gaps occur, from 1 to 57,366; the level share is 28.47 %; 2.1 % of terms are forced level (l <= d^2); 7 terms do not decompose.

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.