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.
n=4: between iv={1,2,...,16} {2,8}U{3,5,6,10,12,16} provides peak values smaller than or equal with 16, so a(4) = 10 = A087256(4)+4
b:= proc(n) option remember; `if`(n=1, 1,
max(n, b(`if`(n::even, n/2, 3*n+1))))
end:
a:= proc(n) option remember; local t; t:=2^n;
add(`if`(b(i)<=t, 1, 0), i=1..t)
end:
seq(a(n), n=0..20); # Alois P. Heinz, Sep 26 2024
c[x_]:=c[x]=(1-Mod[x, 2])*(x/2)+Mod[x, 2]*(3*x+1);c[1]=1; fpl[x_]:=FixedPointList[c, x]; {$RecursionLimit=1000;m=0}; Table[Print[{xm-1, m}];m=0; Do[If[ !Greater[Max[fpl[n]], 2^xm], m=m+1], {n, 1, 2^xm}], {xm, 1, 30}]
Collatz[n_] := NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, # > 1 &]; Table[Length[Select[Range[x=2^n], Max[Collatz[#]] <= x &]], {n,0,10}] (* T. D. Noe, Apr 29 2013 *)
a(4)=3 since 3 is the least number such that largest member of Collatz(3 x + 1) trajectory of 3 is 2^4 = 16.
a225124 = (+ 1) . fromJust . (`elemIndex` a025586_list) . a000079 -- Reinhard Zumkeller, Apr 30 2013
Coll[n_]:=NestWhileList[If[EvenQ[#],#/2,3*#+1]&,n,#>1 &]; t={}; Do[i=1; While[Max[Coll[i]] != 2^n, i++]; AppendTo[t, i], {n,0,25}]; t
n=4: between iv={1,2,...,16} {7,9,11,13,14,15} provides
peak values larger than 16, so a[4]=6.
c[x_]:=c[x]=(1-Mod[x, 2])*(x/2)+Mod[x, 2]*(3*x+1);c[1]=1; fpl[x_]:=FixedPointList[c, x]; {$RecursionLimit=1000;m=0}; Table[Print[{xm-1, m}];m=0; Do[If[Greater[Max[fpl[n]], 2^xm], m=m+1], {n, 1, 2^xm}], {xm, 1, 30}]