[seqfan] Re: Xmas-challenge

[Peter Luschny]

Rob Pratt> This time I used the integer linear programming
solver in SAS to solve an optimal subtour problem.

Rob, thanks for your very informative contribution (in A337125).

- A tightening of the definition is to look for the lexicographically
earliest divisor chain. A more efficient algorithm without
integer linear programming would be desirable. I have set up
the A339491 for this.

- The problem leads to an extension of the tau function.
Let S(n, k) = divisors(k) union {k*j : j = 2..floor(n/k)}.

                                       {1}
                                {1, 2}, {1, 2}
                         {1, 2, 3}, {1, 2}, {1, 3}
            {1, 2, 3, 4}, {1, 2, 4}, {1, 3}, {1, 2, 4}
      {1, 2, 3, 4, 5}, {1, 2, 4}, {1, 3}, {1, 2, 4}, {1, 5}
{1,2,3,4,5,6}, {1,2,4,6}, {1,3,6}, {1,2,4}, {1,5}, {1,2,3,6}

A path in the divisor graph of {1,..,n} is only valid if the
elements of the path p(k-1) are in S(n, p(k)), for k = 2..n.

If one looks at the cardinality of the sets S(n, k), one finds
that the right diagonal gives the natural numbers and the left
diagonal the tau function. I think it's worth taking a closer
look at this. For that, I have set up the triangle A339492.

- We can evaluate the triangle above not only by counting
the elements, but also by forming their sum or product.
And since today is Sylvester's Day, I will also add these
two triangles -- they are now A339496 and A339489.

I want to thank everyone who participated in this end-of-year
challenge.

Bye, Peter