Odd perfect numbers of a special form

What is it about?

It is conjectured that there exists no odd perfect number. In particular, it is conjectured that there exists no odd perfect number of the form $N=p^a (q_1 q_2 \ldots q_k)^{2b}$ with $q_1, q_2, \ldots, q_k$ distinct primes. In this paper, we show that $k\leq 4b^2+2b+2$ and $N\leq 2^{4^{4b^2+2b+3}}$.

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Why is it important?

Several results have been known on odd perfect numbers of the form $N=p^a (q_1 q_2 \ldots q_k)^{2b}$ with $q_1, q_2, \ldots, q_k$ distinct primes. Many of them concern the impossibility of particular values of $b$. This paper gives a result applicable to any value of $b$.

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