# Number of integral solutions

Given a prime number $p$, find the number of pairs of integers $(a, b)$ such that $p \lt a$, $p \lt b$ and $ab$ is divisible by $(a-p)(b-p)$.

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If $f(p)$ is the number of pairs you're counting, I computed $f(2)=14$, $f(3)=26$, $f(5)=38$, $f(7)=44$, $f(11)=56$, $f(13)=50$, $f(17)=62$, $f(19)=68$, $f(23)=80$. The dip at $13$ is surprising. I plugged that sequence into oeis.org and got back nothing. How did this problem come up? – Dimitrije Kostic Dec 1 '11 at 20:27
The "dip" at $13$ is not surprising, since $14$ has few divisors. – André Nicolas Dec 1 '11 at 20:43
@AndréNicolas: I don't understand your comment. The number of divisors of $14$ is only relevant if one of $a$ or $b$ is $14$. – Dimitrije Kostic Dec 2 '11 at 20:07
From divisors of $p^2+p$ one gets the simplest solutions. – André Nicolas Dec 3 '11 at 3:38