# Integers whose Sigma Value is Cube

## 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 $714$