# 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

- 1974: C. Nelson, D.E. Penney and C. Pomerance:
*714 and 715*(*J. Recr. Math.***Vol. 7**,*no. 2*: 87 – 89) - 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $714$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $110$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $714$