Let me explain... I have $n$ integers, with $k$ different values where $k \leq n$. If I sum together the integers with same values I will get a set of different values frequencies. Now if I sum together the similar frequencies I will get another set of only different frequencies. I want to maximize this set.
My question is: can the max size of the set be bounded by a sub-linear function of $n$?
Thank you,
Michael
(If I understand well your problem)
You have $k$ values with each $f_i$ as a frequence this means $k$ tuples $(a_1,f_1),\ldots,(a_k,f_n)$ such that $f_1+\cdots+f_k=n$ and now you extract from $f_i$ the values $f_i'$ and for each value a frequanece $r_i$ hence a sequance of tuples $(f_1',r_1),\ldots,(f'_d,r_d)$ with $r_1+\cdots+r_d=k$ with $f'_1r_1+\cdots+f'_dr_d=n$
we have $k$ maximal when $f_1=f_2=\cdots=f_k=1$ and in this case $k=n$
we have $d$ maximal when $r_1=r_2=\cdots=r_d=1$ and in this cas $d=k$ but because $f_1,f_2,\cdots,f_d$ must be different integers such that $f_1+\cdots+f_d=n$ then it's clear that ${f_1,\cdots,f_d}$ are at least the first positive integers and hence $1+2+\cdots+d\leq n$ hence $\frac{d(d+1)}{2}\leq n$
finally we have $$d\leq \frac{\sqrt{8n+1}-1}{2}$$