Exercise 4.2.4 (Smallest integer $m$ such that $\sigma(m) = \sigma(n)$ for some $n
e m$)

Question

Find the smallest positive integer $m$ for which there is another positive integer $n 
e m$ such that $\sigma(m) = \sigma(n)$.

Richard Ganaye · Accepted Answer

\begin{proof} 
Some values of $\sigma$:
$$
\begin{array}{c|cccccccccccc}
	m		& 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \
\sigma(m)   		&1& 3& 4& 7& 6& 12& 8& 15& 13& 18& 12& 28
\end{array}
$$
We note that $\sigma(6) = \sigma(11) = 12$.
If $n >6$, $\sigma(n) \geq n+1 >7$, thus the integers $1, 3, 4, 7, 6$ have exactly one antecedent, respectively $1,2,3,4,5$.

So the smallest positive integer $m$ for which there is another positive integer $n 
e m$ such that $\sigma(m) = \sigma(n)$ is
$$m = 6.$$
\end{proof}

Exercise 4.2.4 (Smallest integer $m$ such that $\sigma(m) = \sigma(n)$ for some $n \ne m$)

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Exercise 4.2.4 (Smallest integer $m$ such that $\sigma(m) = \sigma(n)$ for some $n \ne m$)

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