new sequences related to partitions

[Edwin Clark]

> 
> 1-A. Let a,b,c,...k be an arbitrary partition of n. 
> let ftau(n) = tau(a) +tau(b)+...tau(k).
> a(n) = maximum value of ftau(n).

If I understand this correctly then a(n) = n for all n.
I assume that tau(a) is the number of positive divisors of a. Then it is
easy to see that tau(a) <=a. So the partition that gives the maximum value
of ftau(n) is n = 1 + 1 + ... + 1. Since tau(1) = 1, we have a(n) = n.

Also if tau is replaced by sigma and max is replaced by min you also get
a(n) = n.
> 
> 1-B
> a(n) = minimum value of ftau(n).
>  If n can be partioned into all primes and a 1 then
> that would give the minimum ftau(n).
> 
> 2. Similarly sq(n) can be defined as a^2 +b^2 ...
> +k^2. and a(n) =  maximum sq(n).
> 
This is a(n)  = n^2

And if you use min instead you get a(n) = n.

> 3. fsigma(n) = sigma(a) + sigma(b) +...+ sigma(k).
> a(n) = maximum value of fsigma(n).
> More such functions can be defined. And the same can
> be extended for  the minimum values.
> 
> thanks
> regards
> amarnath
> 
> 
> NJAS
> 

------------------------------------------------------------
    W. Edwin Clark, Math Dept, University of South Florida,
           http://www.math.usf.edu/~eclark/
------------------------------------------------------------