Integers whose Ratio between Divisor Sum and Phi is Square

Theorem

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

$1, 14, 30, 105, 248, 264, 418, 714, 1485, 3080, \ldots$

This sequence is A293391 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

Examples

Example: $714$

\(\ds \map \phi {714}\)

\(=\)

\(\ds 192\)

$\phi$ of $714$

\(\ds \map {\sigma_1} {714}\)

\(=\)

\(\ds 1728\)

$\sigma_1$ of $714$

\(\ds \map {\sigma_1} {714} / \map \phi {714}\)

\(=\)

\(\ds 1728 / 192\)

\(\ds \)

\(=\)

\(\ds 9\)

\(\ds \)

\(=\)

\(\ds 3^2\)

$\blacksquare$