ranks of (i*j mod n) over R

[Max]

Hi!

The following sequence may deserve to get into the EIS.

For every positive integer n consider nxn matrix with entries (i*j mod 
n), where i,j=0..n-1.
Let a_n be the rank of nth matrix over R (real numbers). Beginning of 
the sequence {a_n} is

0, 1, 2, 3, 3, 5, 4, 6, 6, 7, 6, 10, 7, 9, 10, 11, 9, 13, 10, 14, 13, 
13, 12, 18, 14, 15, 16, 18, 15, 21, 16, 20, 19, 19, 20, 25, 19, 21, 22, 
26, 21, 27, 22, 26, 27, 25, 24, 32, 26, 29, ...

There are two issues:
1) It looks plausible that for every prime p>=3, a_p=(p+1)/2 but I don't 
know the proof.
2) Is there a closed formula for a_n where n is composite?

Regards,
Max