what is the minimum of this sequence

[yogesh]

Hi,

I encountered the following sequence:

Given a positive integer k, consider f(i,k)=a(i,k)/(floor(i/k)), for
all integers i>k-1

where we define a(1,k)=1 and a(i,k) for i>1 by the recursion

a(i,k)=(\sum_{j=1}^{min(i,k)-1} q(i,j)a(j,k)) + i,

where

q(i,j)=(i+1)/(j+1) - 2 when j+1 divides i+1,

=floor((i+1)/(j+1)) when j+1 does not divide i+1.

For instance with k=6 we have,

a(1,6)=1, a(2,6)=3, a(3,6)=6, a(4,6)=15, a(5,6)=27, a(6,6)=63,
a(7,6)=57, a(8,6)=69, a(9,6)=60, a(10,6)=93, a(11,6)=57, a(12,6)=132,
a(13,6)=132, a(14,6)=117, a(15,6)=147, a(16,6)=162, a(17,6)=132 and so
on.

The corresponding f(i,6) sequence for i>5 is obtained by dividing each
term by floor(i/k). Hence

f(6,6)=63, f(7,6)=57, f(8,6)=69, f(9,6)=60, f(10,6)=93, f(11,6)=57,
f(12,6)=66, f(13,6)=66, f(14,6)=58.5, f(15,6)=73.5, f(16,6)=81,
f(17,6)=66 and so on....clearly not monotonic

Now, the question:

What is the minimum over all i>k-1 of f(i,k) for a given k? What is
the corresponding value of the minimizing i? I am interested in values
of k upto 20.

Have such sequences been studied before? If so, relevant references
would be of much help.

In my investigation thus far, I have been able to show that

min(f(i,k)) over all i>k-1 is given by the min(f(i,k)) over all
k-1<i<k+b(k), where b(k) is the LCM(1,2,....,k). However, the search
space is still large as LCM(1,2,....,k) grows very fast with k. Any
ideas on narrowing the search space further, say to a linear in k?
Computer evaluations suggest that this might be the case (probably
searching over k-1<i<2k is sufficient), but I have been unable to
prove this.