Except for Eisenstein series having the divisor functions as their Fourier coefficients, is there any other full integral weight modular form (of some level, preferably full) having arithmetic functions as their Fourier coefficients. 


More to the point, my question is, apart from the relations you obtain between $\sigma\_3, \sigma\_5$ and $\sigma\_7$ are there any applications of full integral weight modular forms (preferably cusp forms) to elementary number theory.



Applications of full integral weight modular forms in elementary number theory
Perhaps this is "out of bounds" given the phrasing of the question, but those Eisenstein series you mention don't just have divisor sums as coefficients - the constant term is a special value of the Riemann zeta function.


This implies all sorts of neat stuff. The various relations between divisor sums that you mention come with relations between zeta values. These give very nice congruences, in particular. 


You can take this to reasoning pretty far to deduce things like $p$-adic interpolation of zeta values a la Kubota-Leopoldt from the much simpler interpolation properties of the divisor sum functions. This was done by Serre in the 1973 paper ("Formes modulaires et fontiones zeta $p$-adiques") that gave birth to the theory of $p$-adic modular forms.


