Let $\sigma(n)$ be the divisor sum of $n$: $$ \sigma(n) = \sum_{d|n} d. $$ I was interested in the parity of $\sigma(n)$ and tried to check whether $\sigma(n)$ is unexpected often even for odd $n$ and vice versa. No result.
Thus I conjectured that $$ \lim_{x\longrightarrow\infty} \dfrac{|\{1:n\leq x,~ Parity(n) \not = Parity(\sigma(n)) \}|}{|\{1:n\leq x, ~Parity(n) = Parity(\sigma(n)) \}|} = 1.$$
I checked this fraction for $x$ up to $100000$ and the conjecture seems to be true. Here are some values:
x Fraction
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10,000 1.02799
50,000 1.01264
100,000 1.00898
However I am not able to prove it. Can you prove that $\sigma(n)$ does not prefer any parity?