Integers whose Sigma Value is Cube

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Theorem

The following positive integers are those whose $\sigma$ value is a cube:

$1, 7, 110, 714, \ldots$

Interestingly, this sequence cannot be found anywhere on the internet, not even on On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Proof

\(\displaystyle \map \sigma 1\) \(=\) \(\, \displaystyle 1 \, \) \(\, \displaystyle =\, \) \(\displaystyle 1^3\) $\sigma$ of $1$
\(\displaystyle \map \sigma 7\) \(=\) \(\, \displaystyle 8 \, \) \(\, \displaystyle =\, \) \(\displaystyle 2^3\) Sigma Function of Prime Number
\(\displaystyle \map \sigma {110}\) \(=\) \(\, \displaystyle 216 \, \) \(\, \displaystyle =\, \) \(\displaystyle 6^3\) $\sigma$ of $110$
\(\displaystyle \map \sigma {714}\) \(=\) \(\, \displaystyle 1728 \, \) \(\, \displaystyle =\, \) \(\displaystyle 12^3\) $\sigma$ of $714$



Sources