Problem Archive

Let $P(m,n)$ be the number of distinct terms in an $m\times n$ multiplication table.

For example, a $3\times 4$ multiplication table looks like this:

$\times$	1	2	3	4
1	1	2	3	4
2	2	4	6	8
3	3	6	9	12

There are $8$ distinct terms $\{1,2,3,4,6,8,9,12\}$, therefore $P(3,4) = 8$.

You are given that:
$P(64,64) = 1263$,
$P(12,345) = 1998$, and
$P(32,10^{15}) = 13826382602124302$.

Find $P(64,10^{16})$.