- A ramp is a sequence of three different positive integers $𝑎,\ 𝑏,\ 𝑐$ such that $𝑎$ is a factor of $𝑏$ and $𝑏$ is a factor of $𝑐$. For every prime number $𝑝$ and every positive integer $𝑛$, determine with proof whether $p^n$ can be expressed as the sum of a ramp.
We observe that if $a,b,c$ constitute a ramp, then we may write it as $a,\alpha a, \alpha \beta a$, where $\alpha$ and $\beta$ are positive integers $>1$. So the sum of the ramp may be written:
$$s(a,b,c)=a(1+\alpha+\alpha \beta)$$
Then, if for some prime $p$ and exponent $n$ $s(a,b,c)=p^n$, we have $a \vert p^n$, so for some $k$ such that $n\ge k\ge 0$ we have $a=p^k$ then $(1+\alpha+\alpha \beta)=p^{n-k}$
The smallest $n$ such that $s(a,b,c)=p^n$ is the smallest $n$ such that $(1+\alpha+\alpha \beta)=p^n$. Given the conditions defining a ramp: $1+\alpha+\alpha \beta \ge 7$, so for any $p\ge 7$ put $\alpha=2$ and $\beta=\frac{p-3}{2}$. Then $1+\alpha+\alpha \beta=p$, and so for every prime $p\ge 7$, and every positive integer $n$ there is a ramp such that $s(a,b,c)=p^n$
That leaves the cases where $p=2$, $p=3$ and $p=5$ to examine.
If $p=5$, $p$ cannot be writen as the sum of a ramp. but $p^2=25$ can as if $\alpha=2, \ \beta=11$ we have $1+\alpha+\alpha \beta=p^2$ and so for $p=5$, $p^n$ can be written as the sum of a ramp for all $n\ge 2$ and no other.
A similar argument applies when $p=3$ as $p^2=9$ as $1+2+2.3=9$
In the case where $p=2$, by trial and error we find the minimum exponent $n=4$
Summary
$p^n$ can be written as a ramp for all primes $\ge 7$ and every positive exponent.
$5^n$ can be written as a ramp for every positive exponent $n\ge 2$ and no others.
$3^n$ can be written as a ramp for every positive exponent $n\ge 2$ and no others.
$2^n$ can be written as a ramp for every positive exponent $n\ge 3$ and no others.
