new sequences related to partitions

[Rob Pratt]

> 2. Similarly sq(n) can be defined as a^2 +b^2 ...
> +k^2. and a(n) =  maximum sq(n).

Unless I'm misinterpreting something, a(n) = n^2 using the trivial
partition of n into one part of size n.  Proof: if the partition has at
least two parts, say u and v, then combine them into a single part
w = u + v; since w^2 = u^2 + 2 u v + v^2 > u^2 + v^2, the objective is
increased.

Similarly, changing maximum to minimum yields a(n) = n using the partition
of n into n parts of size 1.

Rob Pratt
Department of Operations Research
The University of North Carolina at Chapel Hill

rpratt at email.unc.edu

http://www.unc.edu/~rpratt/