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Number Theory

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Moderate

Properties of Sum of Divisors Function

NUMTH-$OKQLE

Denote by $\sigma(n)$ the sum of divisors function of a natural number $n$. Which of the following statements are always true?

Check ALL that apply.

A

$\sigma(n)=\sigma(n+1)$ for all $n$.

B

$\sigma(m)=\sigma(n)$ implies $m=n$.

C

$\sigma(p)=\sigma(p^2)$ for any prime $p$.

D

Let $d(n)$ be the number of divisors function of a natural number $n$. Then $d(n)\leq \sigma(n)\leq nd(n)$ for all $n$.

E

Let $\sigma_{-1}(n)$ be the sum of reciprocal of divisors of $n$, i.e. $\sigma_{-1}(n)=\sum_{d|n} d^{-1}$. Then $n\sigma_{-1}(n)=\sigma(n)$.