Can an odd perfect number be divisible by $105$?

Solution 1:

I did look at the link, but of course the first thing I thought of was Euler's form of an odd perfect number from which this follows immediately. Euler proved that the prime decomposition $n=q^\alpha p_1^{2e_1}\cdots p_m^{2e_m}$ where $q\equiv \alpha\equiv 1\operatorname{mod} 4$, i.e. a perfect square times one extra prime to a power.

Taking this into account, if $n$ is divisible by $3, 5,$ and $7$, then $3$ and $7$ must occur to at least the power $2$.

The abundancy index $I(n)=\frac{\sigma(n)}{n}$ gets larger if you increase powers or number of primes (i.e. for any $k>1$, $I(kn)>I(n)$, see Laatsch's article).

Thus $2=I(n)>I(5\cdot 3^2\cdot 7^2)=\frac{1+5}{5}\cdot\frac{1+3+3^2}{3^2}\cdot\frac{1+7+7^2}{7^2}>2$, a contradiction.