On primes $p$ with $\sigma (p^ \alpha )=m^ 2$

Abstract:

A. Takaku proved that for odd $\alpha \geqslant 3,\sigma ({p^\alpha }) = {m^2},p$ being a prime, implies that $p < {2^{{2^{\alpha + 1}}}}$. In this paper we extend this result to include almost all even integers $\alpha$.