Explanations, extensions requested

[Jack Brennen]

> Can someone explain to me how A000140 and A005184 are generated, and possibly give
> some indication of their significance?
>  
> Also, I would like lists of elements of A051532 and A003273 up to 1001, if possible.

If I'm not mistaken, A005184 consists of each odd number N whose Collatz sequence contains
a higher multiple of N.  Collatz:  a[n+1] = a[n]/2 if a[n] is even, 3*a[n]+1 if a[n] is odd.

For instance, the number 31 is the first element of A005184; its Collatz sequence:

    31, 94, 47, 142, 71, 214, 107, 322, 161, 484, 242, 121, 364, 182, 91, 274, 137,
    412, 206, 103, 310 ...

310 is a multiple of 31, so the number 31 is "self-contained" ...

I don't see any shortcuts for finding such self-contained numbers other than brute-force
search; note though that N can't be divisible by 3, so we only need to check N of the
form 6k+/-1.  A real simple PARI/GP script to do it:

    m=5;d=2;while(1,n=(3*m+1)\2;until(n==1,n=if(n%2,3*n+1,n\2);
                                           if(n%m==0,print(m," ",n);break));m+=d;d=6-d)

The output shows the self-contained numbers along with the higher multiple which
is found in the Collatz sequence:

31 310
83 2158
293 4102
347 4858
671 29524
19151 1417174

If I were going to try to extend the sequence beyond what is shown in the EIS,
I'd use a native language like C -- PARI/GP is just too slow...