A005245 conjecture

[David Wilson]

Let a(n) = A005245(n) = complexity of n = smallest number of 1's needed to 
produce n using operators + and *.

We can work out

[1] a(n) =
        1, if n = 1
        min(MIN({a(x)+a(y): x + y = n, x <= y < n}), MIN({a(x)+a(y): xy = n, 
x <= y < n})), otherwise.

Well, I did some experimenting, and it looks like

[2]  MIN({a(x)+a(y): x + y = n, x <= y < n}) = 1+a(n-1)

that is, the minimum is always attained when x = 1 and y = n-1.  If this is 
true, then the definition can be rewritten

[3] a(n) =
        1, if n = 1
        min(1+a(n-1), MIN({a(x)+a(y): xy = n, x <= y < n})), otherwise.

You gotta admit this is much easier to compute than the [1].

I did a stupid implementation of [3] that ran surprisingly quickly.

Conjecture [2] would be nice to prove, but I've had no luck.