Some new results on odd perfect numbers

Abstract

If m is a multiply perfect number (σ(m) = tm for some integer t), we ask if there is a prime p with (FORMULA PRESENTED)We prove that the only multiply perfect numbers with this property are the even perfect numbers and 672. Hence we settle a problem raised by Suryanarayana who asked if odd perfect numbers necessarily had such a prime factor. The methods of the proof allow us also to say something about odd solutions to the equation σ(σ(n)) = 2n. © 1975 Pacific Journal of Mathematics Manufactured and first issued in Japan.