Is it conjectured that there are no odd multi-perfect numbers?

Question

It is conjectured that there is no odd perfect number.

But is there a stronger conjecture that there are no odd multi-perfect numbers ? Wikipedia shows a useful link, but my conjecture is not mentioned. Here is the link :

http://wwwhomes.uni-bielefeld.de/achim/mpn.html

The site claims that there are infinite many even perfect numbers, although the mersenne-prime-conjecture is still open.

• Does anyone know if there are proofs that all multi-perfect numbers with abundancy $3$ to $6$ have been discovered ?
• Is it conjectured that no odd multi-perfect numbers exist ?

Answer 1

Regarding your question in the title, there is already an answer to a related MO question here.

I hereby reproduce the answer from the referenced MO question here:

The earliest reference seems to be from 1966: E.A. Bugulov, On the question of the existence of odd multiperfect numbers, Kabardino-Balkarskaya State University Učen. Zap. 30 (1966) 9-19. [I could not find this article online.]

Bugulov showed that an odd multiperfect number must have at least 11 distinct prime divisors, more precisely, odd $k$-perfect numbers contain at least $\omega$ distinct prime factors, where $(k,\omega) =$ (3, 11), (4,21), (5, 54),... This result is discussed and improved by Shigeru Nakamura, On k-perfect numbers, Journal of the Tokyo University of Mercantile Marine (Natural Sciences), 33 (1982) 43–50. [Listed here, but not available online.]