decompwlj 3D

Numbers whose base-2 and base-10 expansions have the same digit sum

A037308 on the OEIS · family digit rule

Weight–level plate of Numbers whose base-2 and base-10 expansions have the 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 viewerA037308 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,995
Level class, k > L21,301 · 21.30 %
Weight class, k ≤ L78,694 · 78.70 %
Ties, k = L4
On the level line L = 17,109
Forced level, l ≤ d²1,965
Range of a(n)0 … 100,602,501
Range of the jump d1 … 3,989,877
Largest weight k, level L100,602,331, 50,300,159

1,406 different gaps occur, from 1 to 3,989,877; the level share is 21.30 %; 2.0 % of terms are forced level (l <= d^2); L = 1 holds 33 % 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.