# Integers whose Ratio between Sigma and Phi is Square

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## Theorem

The sequence of integers whose $\sigma$ value divided by its Euler $\phi$ value is a square begins:

- $1, 30, 105, 248, 264, 714, \ldots$

## Proof

\(\displaystyle \map \phi {714}\) | \(=\) | \(\displaystyle 192\) | Euler Phi Function of 714 | ||||||||||

\(\displaystyle \map \sigma {714}\) | \(=\) | \(\displaystyle 1728\) | Sigma of 714 | ||||||||||

\(\displaystyle \map \sigma {714} / \map \phi {714}\) | \(=\) | \(\displaystyle 1728 / 192\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 9\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 3^2\) |

## 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): $714$