computatoin of min/max functions on partitions

[David Wilson]

This regards A. Murthy's idea of sequences with

a(n) = MIN/MAX(P in Partitions(n), f(P))

In the general case, one computes all the paritions in P applies f(P),
and looks for the MIN/MAX of the values.  For large values of n, the
large number of partitions bogs down computation.

However, Murthy's suggested functions have the special form

f(P) = SUM(k in P, g(k))

(Murthy suggested g(k) = tau(k), g(k) = k^2 and g(k) = sigma(k)
though there are obviously endless choices).    When f is of this form
we can show that

a(0) = 0,  a(n) = MIN/MAX({g(n)} U {a(j)+g(n-j); 1 <= j <= [n/2]}), n >= 1.

If we cache values of a(n) and g(n), this computes very quickly.