a(n), for n >= 2, is smallest positive integer which is consistent with sequence being monotonically increasing and satisfying a(a(n)) = 2n

A007378 on the OEIS · family self-referential

Weight–level plate of a(n), for n >= 2, is smallest positive integer which is consistent with sequence being monotonically increasing and satisfying a(a(n)) = 2n — 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 viewer A007378 on the OEIS

Terms	100,000 (n = 2 … 100,001)
Decomposable (a > 2d)	99,999
Level class, k > L	9,512 · 9.51 %
Weight class, k ≤ L	90,487 · 90.49 %
Ties, k = L	35
On the level line L = 1	6,026
Forced level, l ≤ d²	0
Range of a(n)	3 … 134,466
Range of the jump d	1 … 2
Largest weight k, level L	131,063, 65,535

The gaps are 1 and 2; the level share is 9.51 %; L = 1 holds 63 % 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.