Questions tagged [divisor-sum]
For questions on the divisor sum function and its generalizations.
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Average of divisors of n.
Let n be a natural number and let $f(n)=\frac{\sigma(n)}{d(n)}$ be the arithmetical average of n's divisors. Either prove or give a counterexample that for all natural numbers like n, which are not ...
- 4,991
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Number theory Exercise: $\sum_{d \mid n} \mu(d) d(d) = (-1)^{\omega(n)}$ and $\sum_{d \mid n} \mu(d) \sigma (d)$
for positive integer $n$, how can we show
$$ \sum_{d | n} \mu(d) d(d) = (-1)^{\omega(n)} $$
where $d(n)$ is number of positive divisors of $n$ and $mu(n)$ is $(-1)^{\omega(n)} $ if $n$ is square ...
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Finding $\frac{1}{d_1}+\frac{1}{d_2}+\frac{1}{d_3}+...+\frac{1}{d_k}$
If we assume that $d_1,d_2,d_3,...,d_k$ are the divisors for the positive integer $n$ except $1,n$ if $d_1+d_2+d_3+...+ d_k=72$ then how to find $$\frac{1}{d_1}+\frac{1}{d_2}+\frac{1}{d_3}+...+\frac{...
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Inclusion Exclusion and lcm
I would like to show that for any positive integers $d_1, \dots, d_r$ one has
$$
\sum_{i=1}^r (-1)^{i+1}\biggl( \sum_{1\leq k_1 < \dots < k_i \leq r} \gcd(d_{k_1}, \dots , d_{k_i})\biggr) ~\leq~ ...
- 21
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When is the sum of divisors a prime?
Can we efficiently figure out when the sum of divisors of a number can be a prime?
I realized that this can be possible only when the number is expressible as a power of only one prime, e.g. $n = p^\...
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Show that $\sum\nolimits_{d|n} \frac{1}{d} = \frac{\sigma (n)}{n}$ for every positive integer $n$.
Show that $\sum\nolimits_{d|n} \frac{1}{d} = \frac{\sigma (n)}{n}$ for every positive integer $n$.
where $\sigma (n)$ is the sum of all the divisors of $n$
and $\sum\nolimits_{d|n} f(d)$ is the ...
- 3,188
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Sum of Positive Divisors: $\sum_{d|n} \mu(n/d)\nu(d)=1$ and $\sum_{d|n} \mu(n/d)\sigma(d)=n$
If $\nu(n)=$ Number of positive divisors of $n,$ $\mu$ is the Möbius function and
$\sigma(n)$ is the sum of positive divisors.
show that;
$\sum\limits_{d|n} \mu(n/d)\nu(d)=1$ for all $n.$
$\sum\...
- 191
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Composite $n$ such that $\sigma(n) \equiv n+1 \pmod{\phi(n)}$
I'm looking for composite $n$ such that
$$\sigma(n)\equiv n+1\pmod{\varphi(n)}$$
Are there only finitely many? Can this be proved?
This is Sloane's A070037 but there's not much information in the ...
- 32.3k
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An approximate relationship between the totient function and sum of divisors
I was playing around with a few of the number theory functions in Mathematica when I found an interesting relationship between some of them. Below I have plotted points with coordinates $x=\dfrac{n\...
- 315
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Sum of divisor ratio inequality
Consider the divisors of $n$, $$d_1 = 1, d_2, d_3, ..., d_r=n$$ in ascending order and $r \equiv r(n)$ is the number of divisors of $n$.
Is there any expression $f(n) < r(n)$ such that,
$$\sum_{k=...
- 1,913
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$\sum_{i=1}^n |\{k \in \mathbb{N} \mid k | i\}|$
What is $\sum_{i=1}^n |\{k \in \mathbb{N} \mid k | i\}|$ asymptotically (as a function of $n$)?
(I'm summing, for each of $1,\dotsc,n$, its number of divisors)
Or at least, what's the best upper ...
- 83
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Proving $ \frac{\sigma(n)}{n} < \frac{n}{\varphi(n)} < \frac{\pi^{2}}{6} \frac{\sigma(n)}{n}$
This is an exercise from Apostol's number theory book. How does, one prove that $$ \frac{\sigma(n)}{n} < \frac{n}{\varphi(n)} < \frac{\pi^{2}}{6} \frac{\sigma(n)}{n} \quad \text{if} \ n \geq 2$$ ...
anonymous