A005245 conjecture

[Max Alekseyev]

On Jan 31, 2008 3:57 AM, David Wilson <davidwwilson at comcast.net> wrote:

> Again, for a = A005245, let
>
>     pmin(n) = MIN({a(x)a(y): xy = n, > x <= y < n})
>     smin(n) = MIN({a(x)+a(y): xy = n, > x <= y < n})
>
> pmin(n) is defined only for composite n, smin(n) for n >= 2.
>
> A recursive definition of a is then
>
>     a(n) = {
>         1; n = 1
>         smin(n); n prime
>         min(smin(n), pmin(n)); n composite
>     }
>
> Even though general smin(n) = 1+a(n-1) has failed us, one could still hope
> for [3] below, that is,
>
>     a(n) = {
>         1; n = 1
>         1+a(n-1); n prime
>         pmin(n); n composite
>     }

This statement looks stronger than [3]. In addition to what is stated
in [3], the above statement claims that pmin(n) <= smin(n) for every
composite n. Or does that follow from [3] and I'm missing some trivial
arguments?

[...]

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

Regards,
Max