decompwlj 3D

Palindromes with even number of digits

A056524 on the OEIS · family digit rule

Weight–level plate of Palindromes with even number of digits
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 viewerA056524 on the OEIS
Terms100,000 (n = 1 … 100,000)
Decomposable (a > 2d)99,993
Level class, k > L94,490 · 94.50 %
Weight class, k ≤ L5,503 · 5.50 %
Ties, k = L0
On the level line L = 15
Forced level, l ≤ d²89,998
Range of a(n)11 … 100,000,000,001
Range of the jump d11 … 90,000,000,002
Largest weight k, level L909,070,909, 411,764,705

11 different gaps occur, from 11 to 90,000,000,002; the level share is 94.50 %; 90.0 % of terms are forced level (l <= d^2); there are no ties; 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.