Integers which are Divisor Sum for 3 Integers

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Theorem

The sequence of integers which are the divisor sum of $3$ different integers begins:

$24, 42, 48, 60, 84, 90, \ldots$

This sequence is A007372 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Proof

For a given $n$, to determine every $m$ such that $\map {\sigma_1} m = n$ can be determined by evaluating the divisor sum of all integers up to $n - 1$.


It is hence noted:

\(\ds 24\) \(=\) \(\ds \map {\sigma_1} {14}\) $\sigma_1$ of $14$
\(\ds \) \(=\) \(\ds \map {\sigma_1} {15}\) $\sigma_1$ of $15$
\(\ds \) \(=\) \(\ds \map {\sigma_1} {23}\) Divisor Sum of Prime Number: $23$ is prime


\(\ds 42\) \(=\) \(\ds \map {\sigma_1} {20}\) $\sigma_1$ of $20$
\(\ds \) \(=\) \(\ds \map {\sigma_1} {26}\) $\sigma_1$ of $26$
\(\ds \) \(=\) \(\ds \map {\sigma_1} {41}\) Divisor Sum of Prime Number: $41$ is prime


\(\ds 48\) \(=\) \(\ds \map {\sigma_1} {33}\) $\sigma_1$ of $33$
\(\ds \) \(=\) \(\ds \map {\sigma_1} {35}\) $\sigma_1$ of $35$
\(\ds \) \(=\) \(\ds \map {\sigma_1} {47}\) Divisor Sum of Prime Number: $47$ is prime


\(\ds 60\) \(=\) \(\ds \map {\sigma_1} {24}\) $\sigma_1$ of $24$
\(\ds \) \(=\) \(\ds \map {\sigma_1} {38}\) $\sigma_1$ of $38$
\(\ds \) \(=\) \(\ds \map {\sigma_1} {59}\) Divisor Sum of Prime Number: $59$ is prime


\(\ds 84\) \(=\) \(\ds \map {\sigma_1} {44}\) $\sigma_1$ of $44$
\(\ds \) \(=\) \(\ds \map {\sigma_1} {65}\) $\sigma_1$ of $65$
\(\ds \) \(=\) \(\ds \map {\sigma_1} {83}\) Divisor Sum of Prime Number: $83$ is prime


\(\ds 90\) \(=\) \(\ds \map {\sigma_1} {40}\) $\sigma_1$ of $40$
\(\ds \) \(=\) \(\ds \map {\sigma_1} {58}\) $\sigma_1$ of $58$
\(\ds \) \(=\) \(\ds \map {\sigma_1} {89}\) Divisor Sum of Prime Number: $89$ is prime

$\blacksquare$


Sources

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