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Dec
8
comment Ratio of consecutive divisors and average
We have $D(x,t) \asymp x \log t / \log x$ for $x \ge t \ge 2$, a result due to Saias. So for fixed $t$, the integers with $F(n)/n \le t$ have zero density.
Dec
8
comment Ratio of consecutive divisors and average
I don't think there is a typo. Because you are looking for a maximum, you can take $\beta_i = \alpha_i$.
Dec
8
revised Ratio of consecutive divisors and average
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Dec
8
revised Ratio of consecutive divisors and average
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Dec
8
comment Ratio of consecutive divisors and average
Yes, $d|n$ implies $d\le n$ in the definition of $F$.
Dec
8
revised Ratio of consecutive divisors and average
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Dec
8
revised Ratio of consecutive divisors and average
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Dec
8
revised Ratio of consecutive divisors and average
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Dec
8
answered Ratio of consecutive divisors and average
Nov
21
comment A function whose fixed points are the primes
Another problem of the same flavor (and just as difficult) is $f(n)=\sigma(n)-1$, where $\sigma(n)$ is the sum of the divisors of $n$.
Feb
15
revised Partitioning the integers $1$ through $n$ so that the product of the elements in one set is equal to the sum of the elements in the other
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Feb
15
revised Partitioning the integers $1$ through $n$ so that the product of the elements in one set is equal to the sum of the elements in the other
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Feb
14
awarded  Editor
Feb
14
revised Partitioning the integers $1$ through $n$ so that the product of the elements in one set is equal to the sum of the elements in the other
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Feb
14
answered Partitioning the integers $1$ through $n$ so that the product of the elements in one set is equal to the sum of the elements in the other
Feb
13
awarded  Supporter
Feb
13
awarded  Teacher
Feb
13
answered Erdos-Kac for sum of divisors