Divisibility in the partition function

Question

There are several formulas for the calculation of the partition function $p(n)$ of an integer $n$. The last has been found by Ken Ono in 2011. My question is: using these formulas is it possible to calculate for how many $n$ the ratio $R=\frac{p(n+2)}{p(n)}$ is an integer number? For example, the $p(6)=11$ and $p(8)=22$, so in this case, $R=2$. Is this the only case? Thanks.

Answer 1

There is the trivial case $p(3)/p(1)=3$, Besides the case you mention there is also $p(9)/p(7)=2.$ There cannot be more, provided one can show that $p(n+2)/p(n)$ is weakly monotone decreasing, as it seems to be numerically; at $n=8$ this ratio is already below $2$ at $n=8$ where it is $21/11.$ I don't know offhand how to show the weak monotonicity.

ADDED: Actually all one needs to show is that $p(n+2)/p(n)<2$ for $n \ge 8$, in order to exclude the possibility of more solutions to $p(n+2)/p(n)$ being an integer, since it can clearly not be $1$.