I recently found OEIS entry A256504 and have been playing around with this sequence a bit. Its definition is:
For a positive integer $n$, find the greatest number of consecutive positive integers (at least 2) which add to $n$. For each of these do the same ... iterate to completion. $F(n)$ is then the total number of integers (including $n$ itself) defined.
And here is an example, $F(24) = 13$:
24
/|\
/ | \
/ | \
7 8 9
/ \ /|\
3 4 / | \
/ \ / | \
1 2 2 3 4
/ \
1 2
Looking at the graph on OEIS these numbers seem to be growing roughly linearly, but I'm not sure how to establish a rigorous bound for it.
A few interesting things I've noticed while checking numbers up to $n = 6\cdot10^6$ (and a few around $10^7$):
- There only seems to be one number where $F(n) > n$: $F(11) = 12$
- There only seem to be five numbers where $F(n) = n$: $1$, $3$, $5$, $6$, $23$.
It appears that the sequence is growing slightly sublinearly.
After $10^6$, the largest ratio $F(n)/n$ I can find is $0.713924$ for $1110609$.
After $2\cdot10^6$, it's $0.712693$ for $2097749$.
After $5\cdot10^6$, it's $0.710524$ for $5570687$.
After $10^7$, it's $0.709625$ for $10240519$.
So my question is: can an upper bound for the sequence's growth be established rigorously and how tight can we make it? If the maximum ratio $\max_{n>n_0}F(n)/n$ is actually decreasing as $n_0$ grows, does it approach a finite constant?
I suppose it might be helpful to have an upper bound on A109814.