Hailstone collatz max sequence length upper bound of $260.5+x^{.43}$?

Question

Let the Collatz function be defined as if $x$ even $c(x)=x/2$, if $x$ odd then $c(x)=3x+1$ over the naturals. Each operation is defined as a step. For example $3$ goes $(3,10,5,16,8,4,2,1)$ and takes 8 steps to reach one.

Can anyone find an example for me where the number of steps for starting natural number $x$ to reach one is greater then $260.5+x^.43$?

Here is a graph with the max number of collatz length and the fitted upper bound

slightly more evidence from a slower function on a bigger set. (function of $460+x^{(1/3)}$)

I conjecture that there is no such number. Is there any other works of this type?

Answer 1

I've searched up to $10^8$ and the closest you get to your bound is at $230631$ which appears in your first graph. Nothing else is even above $240 + x^{.43}$ let alone your bound. $$\{x \in \Bbb{Z}|\quad0<x\le10^8 , \quad f(x) > 240 + x^{.43} \} = \{230631\}$$

As was mentioned in the comments. There are no known results of the type you are asking. I did the search to answer your query for an in-depth search.

#include <stdio.h>
#include <stdlib.h>
#include <math.h>
#define u64 long long unsigned

u64 *c;
u64 N = 100000000LL;
u64 collatz( u64 a )
{
    if( a < N && c[a] ) return c[a];
    u64 t = 1 + collatz( 1&a ? 3*a + 1 : a >> 1 );
    return ( a < N ? c[a] = t : t );
}

int main()
{
    c = calloc( N, sizeof(u64) );
    c[1] = 1;
    for( u64 i = 2; i < N; ++i )
        if( collatz( i ) >= (u64) ceil( 240. + pow( i, 0.43 ) ) ) 
            printf( "%llu\n", i );
}

Answer 2

Just to illustrate, how the (empirically for $a_1<=1 000 000$) tendency and the upper bound given by the OP's proposal diverge in order . That means, they might come near, but after a region of near-matches they diverge because of the different type of functions: by OP it is of order $a_1^C$ with $C$ a constant, while the heuristig suggests a form like $\ln(a_1)^C$. Of course they must diverge from each other.
See the picture: The grey dots indicate the measures of total stopping time for the first $500$ odd numbers $a_1$, the blue line and the larger circles indicate the moving maxima (or: "upper envelope") and the thick red line indicate the trend which is determined by excel. The magenta curve comes by the ansatz from the OP.

If the assumption that the visible curve of the envelope has a logarithmic trend is meaningful, then it is obvious that the nearest values of the OP's assumed trend and the empirical trend is in the region of $a_1 \approx 10^5, N+S \approx 400$ andto search a counterexample in higher $a_1$ should be meaningless.

Update: Using $a_1$ up to $10^7$ it became more suggestive, that a trend ignoring the few leading smallest values in the picture should possibly be linear; a regression (plus a manual shift) gave the according equation $N+S \approx 73.5327 a_1^{0.1532}$