Numbers such that Tau divides Phi divides Sigma

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Theorem

The sequence of integers $n$ with the property that:

$\map \tau n \divides \map \phi n \divides \map \sigma n$

where:

$\divides$ denotes divisibility
$\tau$ denotes the $\tau$ (tau) function: the count of divisors of $n$
$\phi$ denotes the Euler $\phi$ (phi) function: the count of smaller integers coprime to $n$
$\sigma$ denotes the $\sigma$ (sigma) function: the sum of divisors of $n$

begins:

$1, 3, 15, 30, 35, 56, 70, 78, 105, 140, 168, 190, 210, 248, 264, \ldots$

This sequence is A020493 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Proof

By inspection and investigation.

$\blacksquare$


Examples

1

\(\displaystyle \map \tau 1\) \(=\) \(\, \displaystyle 1 \, \) \(\displaystyle \) $\tau$ of $1$
\(\displaystyle \map \phi 1\) \(=\) \(\, \displaystyle 1 \, \) \(\displaystyle \) $\phi$ of $1$
\(\displaystyle \map \sigma 1\) \(=\) \(\, \displaystyle 1 \, \) \(\displaystyle \) $\sigma$ of $1$

$\blacksquare$


3

\(\displaystyle \map \tau 3\) \(=\) \(\, \displaystyle 2 \, \) \(\displaystyle \) $\tau$ of $3$
\(\displaystyle \map \phi 3\) \(=\) \(\, \displaystyle 2 \, \) \(\displaystyle \) $\phi$ of $3$
\(\displaystyle \map \sigma 3\) \(=\) \(\, \displaystyle 4 \, \) \(\, \displaystyle =\, \) \(\displaystyle 2 \times 2\) $\sigma$ of $3$

$\blacksquare$


15

\(\displaystyle \tau \left({15}\right)\) \(=\) \(\, \displaystyle 4 \, \) \(\displaystyle \) $\tau$ of $15$
\(\displaystyle \phi \left({15}\right)\) \(=\) \(\, \displaystyle 8 \, \) \(\, \displaystyle =\, \) \(\displaystyle 2 \times 4\) $\phi$ of $15$
\(\displaystyle \sigma \left({15}\right)\) \(=\) \(\, \displaystyle 24 \, \) \(\, \displaystyle =\, \) \(\displaystyle 3 \times 8\) $\sigma$ of $15$

$\blacksquare$


30

\(\displaystyle \tau \left({30}\right)\) \(=\) \(\, \displaystyle 8 \, \) \(\displaystyle \) $\tau$ of $30$
\(\displaystyle \phi \left({30}\right)\) \(=\) \(\, \displaystyle 8 \, \) \(\displaystyle \) $\phi$ of $30$
\(\displaystyle \sigma \left({30}\right)\) \(=\) \(\, \displaystyle 72 \, \) \(\, \displaystyle =\, \) \(\displaystyle 9 \times 8\) $\sigma$ of $30$

$\blacksquare$


35

\(\displaystyle \tau \left({35}\right)\) \(=\) \(\, \displaystyle 4 \, \) \(\displaystyle \) $\tau$ of $35$
\(\displaystyle \phi \left({35}\right)\) \(=\) \(\, \displaystyle 24 \, \) \(\, \displaystyle =\, \) \(\displaystyle 6 \times 4\) $\phi$ of $35$
\(\displaystyle \sigma \left({35}\right)\) \(=\) \(\, \displaystyle 48 \, \) \(\, \displaystyle =\, \) \(\displaystyle 2 \times 24\) $\sigma$ of $35$

$\blacksquare$


56

\(\displaystyle \tau \left({56}\right)\) \(=\) \(\, \displaystyle 8 \, \) \(\displaystyle \) $\tau$ of $56$
\(\displaystyle \phi \left({56}\right)\) \(=\) \(\, \displaystyle 24 \, \) \(\, \displaystyle =\, \) \(\displaystyle 3 \times 8\) $\phi$ of $56$
\(\displaystyle \sigma \left({56}\right)\) \(=\) \(\, \displaystyle 120 \, \) \(\, \displaystyle =\, \) \(\displaystyle 5 \times 24\) $\sigma$ of $56$

$\blacksquare$


70

\(\displaystyle \map \tau {70}\) \(=\) \(\, \displaystyle 8 \, \) \(\displaystyle \) $\tau$ of $70$
\(\displaystyle \map \phi {70}\) \(=\) \(\, \displaystyle 24 \, \) \(\, \displaystyle =\, \) \(\displaystyle 3 \times 8\) $\phi$ of $70$
\(\displaystyle \map \sigma {70}\) \(=\) \(\, \displaystyle 144 \, \) \(\, \displaystyle =\, \) \(\displaystyle 6 \times 24\) $\sigma$ of $70$

$\blacksquare$


78

\(\displaystyle \map \tau {78}\) \(=\) \(\, \displaystyle 8 \, \) \(\displaystyle \) $\tau$ of $78$
\(\displaystyle \map \phi {78}\) \(=\) \(\, \displaystyle 24 \, \) \(\, \displaystyle =\, \) \(\displaystyle 3 \times 8\) $\phi$ of $78$
\(\displaystyle \map \sigma {78}\) \(=\) \(\, \displaystyle 168 \, \) \(\, \displaystyle =\, \) \(\displaystyle 7 \times 24\) $\sigma$ of $78$

$\blacksquare$


105

\(\displaystyle \map \tau {105}\) \(=\) \(\, \displaystyle 8 \, \) \(\displaystyle \) $\tau$ of $105$
\(\displaystyle \map \phi {105}\) \(=\) \(\, \displaystyle 48 \, \) \(\, \displaystyle =\, \) \(\displaystyle 6 \times 8\) $\phi$ of $105$
\(\displaystyle \map \sigma {105}\) \(=\) \(\, \displaystyle 192 \, \) \(\, \displaystyle =\, \) \(\displaystyle 4 \times 48\) $\sigma$ of $105$

$\blacksquare$


140

\(\displaystyle \tau \left({140}\right)\) \(=\) \(\, \displaystyle 12 \, \) \(\displaystyle \) $\tau$ of $140$
\(\displaystyle \phi \left({140}\right)\) \(=\) \(\, \displaystyle 48 \, \) \(\, \displaystyle =\, \) \(\displaystyle 4 \times 12\) $\phi$ of $140$
\(\displaystyle \sigma \left({140}\right)\) \(=\) \(\, \displaystyle 336 \, \) \(\, \displaystyle =\, \) \(\displaystyle 7 \times 48\) $\sigma$ of $140$

$\blacksquare$


168

\(\displaystyle \tau \left({168}\right)\) \(=\) \(\, \displaystyle 16 \, \) \(\displaystyle \) $\tau$ of $168$
\(\displaystyle \phi \left({168}\right)\) \(=\) \(\, \displaystyle 48 \, \) \(\, \displaystyle =\, \) \(\displaystyle 3 \times 16\) $\phi$ of $168$
\(\displaystyle \sigma \left({168}\right)\) \(=\) \(\, \displaystyle 480 \, \) \(\, \displaystyle =\, \) \(\displaystyle 10 \times 48\) $\sigma$ of $168$

$\blacksquare$


190

\(\displaystyle \tau \left({190}\right)\) \(=\) \(\, \displaystyle 8 \, \) \(\displaystyle \) $\tau$ of $190$
\(\displaystyle \phi \left({190}\right)\) \(=\) \(\, \displaystyle 72 \, \) \(\, \displaystyle =\, \) \(\displaystyle 9 \times 8\) $\phi$ of $190$
\(\displaystyle \sigma \left({190}\right)\) \(=\) \(\, \displaystyle 360 \, \) \(\, \displaystyle =\, \) \(\displaystyle 5 \times 72\) $\sigma$ of $190$

$\blacksquare$


210

\(\displaystyle \tau \left({210}\right)\) \(=\) \(\, \displaystyle 16 \, \) \(\displaystyle \) $\tau$ of $210$
\(\displaystyle \phi \left({210}\right)\) \(=\) \(\, \displaystyle 48 \, \) \(\, \displaystyle =\, \) \(\displaystyle 3 \times 16\) $\phi$ of $210$
\(\displaystyle \sigma \left({210}\right)\) \(=\) \(\, \displaystyle 576 \, \) \(\, \displaystyle =\, \) \(\displaystyle 12 \times 48\) $\sigma$ of $210$

$\blacksquare$


248

\(\displaystyle \tau \left({248}\right)\) \(=\) \(\, \displaystyle 8 \, \) \(\displaystyle \) $\tau$ of $248$
\(\displaystyle \phi \left({248}\right)\) \(=\) \(\, \displaystyle 120 \, \) \(\, \displaystyle =\, \) \(\displaystyle 15 \times 8\) $\phi$ of $248$
\(\displaystyle \sigma \left({248}\right)\) \(=\) \(\, \displaystyle 480 \, \) \(\, \displaystyle =\, \) \(\displaystyle 4 \times 120\) $\sigma$ of $248$

$\blacksquare$


264

\(\displaystyle \tau \left({264}\right)\) \(=\) \(\, \displaystyle 16 \, \) \(\displaystyle \) $\tau$ of $264$
\(\displaystyle \phi \left({264}\right)\) \(=\) \(\, \displaystyle 80 \, \) \(\, \displaystyle =\, \) \(\displaystyle 5 \times 16\) $\phi$ of $264$
\(\displaystyle \sigma \left({264}\right)\) \(=\) \(\, \displaystyle 720 \, \) \(\, \displaystyle =\, \) \(\displaystyle 9 \times 80\) $\sigma$ of $264$

$\blacksquare$


357

\(\displaystyle \tau \left({357}\right)\) \(=\) \(\, \displaystyle 8 \, \) \(\displaystyle \) $\tau$ of $357$
\(\displaystyle \phi \left({357}\right)\) \(=\) \(\, \displaystyle 192 \, \) \(\, \displaystyle =\, \) \(\displaystyle 24 \times 8\) $\phi$ of $357$
\(\displaystyle \sigma \left({357}\right)\) \(=\) \(\, \displaystyle 576 \, \) \(\, \displaystyle =\, \) \(\displaystyle 3 \times 192\) $\sigma$ of $357$

$\blacksquare$


420

\(\displaystyle \tau \left({420}\right)\) \(=\) \(\, \displaystyle 24 \, \) \(\displaystyle \) $\tau$ of $420$
\(\displaystyle \phi \left({420}\right)\) \(=\) \(\, \displaystyle 96 \, \) \(\, \displaystyle =\, \) \(\displaystyle 4 \times 24\) $\phi$ of $420$
\(\displaystyle \sigma \left({420}\right)\) \(=\) \(\, \displaystyle 1344 \, \) \(\, \displaystyle =\, \) \(\displaystyle 14 \times 96\) $\sigma$ of $420$

$\blacksquare$


570

\(\displaystyle \tau \left({570}\right)\) \(=\) \(\, \displaystyle 16 \, \) \(\displaystyle \) $\tau$ of $570$
\(\displaystyle \phi \left({570}\right)\) \(=\) \(\, \displaystyle 144 \, \) \(\, \displaystyle =\, \) \(\displaystyle 9 \times 16\) $\phi$ of $570$
\(\displaystyle \sigma \left({570}\right)\) \(=\) \(\, \displaystyle 1440 \, \) \(\, \displaystyle =\, \) \(\displaystyle 10 \times 144\) $\sigma$ of $570$

$\blacksquare$


616

\(\displaystyle \tau \left({616}\right)\) \(=\) \(\, \displaystyle 16 \, \) \(\displaystyle \) $\tau$ of $616$
\(\displaystyle \phi \left({616}\right)\) \(=\) \(\, \displaystyle 240 \, \) \(\, \displaystyle =\, \) \(\displaystyle 15 \times 16\) $\phi$ of $616$
\(\displaystyle \sigma \left({616}\right)\) \(=\) \(\, \displaystyle 1440 \, \) \(\, \displaystyle =\, \) \(\displaystyle 6 \times 240\) $\sigma$ of $616$

$\blacksquare$


630

\(\displaystyle \map \tau {630}\) \(=\) \(\, \displaystyle 24 \, \) \(\displaystyle \) $\tau$ of $630$
\(\displaystyle \map \phi {630}\) \(=\) \(\, \displaystyle 144 \, \) \(\, \displaystyle =\, \) \(\displaystyle 6 \times 24\) $\phi$ of $630$
\(\displaystyle \map \sigma {630}\) \(=\) \(\, \displaystyle 1872 \, \) \(\, \displaystyle =\, \) \(\displaystyle 13 \times 144\) $\sigma$ of $630$

$\blacksquare$


Sources