This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A364803 #10 Aug 25 2023 12:44:53
%S A364803 0,4,5,6,7,10,14,23,46,117,442,3006,47983,2839934,918486751,
%T A364803 3769839124330
%N A364803 Smallest number that reaches a fixed point after n iterations of the map x -> A022290(x).
%C A364803 a(n) is the smallest number k such that A364801(k) = n.
%t A364803 f[n_] := f[n] = Module[{d = IntegerDigits[n, 2], nd}, nd = Length[d]; Total[d * Fibonacci[Range[nd + 1, 2, -1]]]]; (* A022290 *)
%t A364803 iternum[n_] := -2 + Length@ FixedPointList[f, n]; (* A364801 *)
%t A364803 seq[kmax_] := Module[{s = {}, imax = -1, i}, Do[i = iternum[k]; If[i > imax, imax = i; AppendTo[s, k]], {k, 0, kmax}]; s]
%t A364803 seq[10^6]
%o A364803 (PARI) f(n) = {my(b = binary(n), nb = #b); sum(i = 1, nb, b[i] * fibonacci(nb - i + 2)); } \\ A022290
%o A364803 iternum(n) = if(n < 4, 0, iternum(f(n)) + 1); \\ A364801
%o A364803 lista(kmax) = {my(imax = -1, i1); for(k = 0, kmax, i = iternum(k); if(i > imax, imax = i; print1(k, ", ")));}
%Y A364803 Cf. A022290, A364801.
%Y A364803 Similar sequences: A007755, A364802.
%K A364803 nonn,base,more
%O A364803 0,2
%A A364803 _Amiram Eldar_, Aug 08 2023
%E A364803 a(15) from _Martin Ehrenstein_, Aug 25 2023