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We shall solve the simultaneous equations $\sigma(2^a)=p^{f_1}q^{g_1}, \sigma(3^b)=p^{f_2}q^{g_2}, \sigma(5^c)=p^{f_3}q^{g_3}$ with $p, q$ distinct primes.

On the simultaneous equations $\sigma(2^a)=p^{f_1}q^{g_1}$, $\sigma(3^b)=p^{f_2}q^{g_2}$, $\sigma(5^c)=p^{f_3}q^{g_3}$