Numbers such that Divisor Count divides Phi divides Divisor Sum/Examples
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Examples of Numbers such that Divisor Count divides Phi divides Divisor Sum
1
| \(\ds \map {\sigma_0} 1\) | \(=\) | \(\, \ds 1 \, \) | \(\ds \) | $\sigma_0$ of $1$ | ||||||||||
| \(\ds \map \phi 1\) | \(=\) | \(\, \ds 1 \, \) | \(\ds \) | $\phi$ of $1$ | ||||||||||
| \(\ds \map {\sigma_1} 1\) | \(=\) | \(\, \ds 1 \, \) | \(\ds \) | $\sigma_1$ of $1$ |
$\blacksquare$
3
| \(\ds \map {\sigma_0} 3\) | \(=\) | \(\, \ds 2 \, \) | \(\ds \) | $\sigma_0$ of $3$ | ||||||||||
| \(\ds \map \phi 3\) | \(=\) | \(\, \ds 2 \, \) | \(\ds \) | $\phi$ of $3$ | ||||||||||
| \(\ds \map {\sigma_1} 3\) | \(=\) | \(\, \ds 4 \, \) | \(\, \ds = \, \) | \(\ds 2 \times 2\) | $\sigma_1$ of $3$ |
$\blacksquare$
15
| \(\ds \map {\sigma_0} {15}\) | \(=\) | \(\, \ds 4 \, \) | \(\ds \) | $\sigma_0$ of $15$ | ||||||||||
| \(\ds \map \phi {15}\) | \(=\) | \(\, \ds 8 \, \) | \(\, \ds = \, \) | \(\ds 2 \times 4\) | $\phi$ of $15$ | |||||||||
| \(\ds \map {\sigma_1} {15}\) | \(=\) | \(\, \ds 24 \, \) | \(\, \ds = \, \) | \(\ds 3 \times 8\) | $\sigma_1$ of $15$ |
$\blacksquare$
30
| \(\ds \map {\sigma_0} {30}\) | \(=\) | \(\, \ds 8 \, \) | \(\ds \) | $\sigma_0$ of $30$ | ||||||||||
| \(\ds \map \phi {30}\) | \(=\) | \(\, \ds 8 \, \) | \(\ds \) | $\phi$ of $30$ | ||||||||||
| \(\ds \map {\sigma_1} {30}\) | \(=\) | \(\, \ds 72 \, \) | \(\, \ds = \, \) | \(\ds 9 \times 8\) | $\sigma_1$ of $30$ |
$\blacksquare$
35
| \(\ds \map {\sigma_0} {35}\) | \(=\) | \(\, \ds 4 \, \) | \(\ds \) | $\sigma_0$ of $35$ | ||||||||||
| \(\ds \map \phi {35}\) | \(=\) | \(\, \ds 24 \, \) | \(\, \ds = \, \) | \(\ds 6 \times 4\) | $\phi$ of $35$ | |||||||||
| \(\ds \map {\sigma_1} {35}\) | \(=\) | \(\, \ds 48 \, \) | \(\, \ds = \, \) | \(\ds 2 \times 24\) | $\sigma_1$ of $35$ |
$\blacksquare$
56
| \(\ds \map {\sigma_0} {56}\) | \(=\) | \(\, \ds 8 \, \) | \(\ds \) | $\sigma_0$ of $56$ | ||||||||||
| \(\ds \map \phi {56}\) | \(=\) | \(\, \ds 24 \, \) | \(\, \ds = \, \) | \(\ds 3 \times 8\) | $\phi$ of $56$ | |||||||||
| \(\ds \map {\sigma_1} {56}\) | \(=\) | \(\, \ds 120 \, \) | \(\, \ds = \, \) | \(\ds 5 \times 24\) | $\sigma_1$ of $56$ |
$\blacksquare$
70
| \(\ds \map {\sigma_0} {70}\) | \(=\) | \(\, \ds 8 \, \) | \(\ds \) | $\sigma_0$ of $70$ | ||||||||||
| \(\ds \map \phi {70}\) | \(=\) | \(\, \ds 24 \, \) | \(\, \ds = \, \) | \(\ds 3 \times 8\) | $\phi$ of $70$ | |||||||||
| \(\ds \map {\sigma_1} {70}\) | \(=\) | \(\, \ds 144 \, \) | \(\, \ds = \, \) | \(\ds 6 \times 24\) | $\sigma_1$ of $70$ |
$\blacksquare$
78
| \(\ds \map {\sigma_0} {78}\) | \(=\) | \(\, \ds 8 \, \) | \(\ds \) | $\sigma_0$ of $78$ | ||||||||||
| \(\ds \map \phi {78}\) | \(=\) | \(\, \ds 24 \, \) | \(\, \ds = \, \) | \(\ds 3 \times 8\) | $\phi$ of $78$ | |||||||||
| \(\ds \map {\sigma_1} {78}\) | \(=\) | \(\, \ds 168 \, \) | \(\, \ds = \, \) | \(\ds 7 \times 24\) | $\sigma_1$ of $78$ |
$\blacksquare$
105
| \(\ds \map {\sigma_0} {105}\) | \(=\) | \(\, \ds 8 \, \) | \(\ds \) | $\sigma_0$ of $105$ | ||||||||||
| \(\ds \map \phi {105}\) | \(=\) | \(\, \ds 48 \, \) | \(\, \ds = \, \) | \(\ds 6 \times 8\) | $\phi$ of $105$ | |||||||||
| \(\ds \map {\sigma_1} {105}\) | \(=\) | \(\, \ds 192 \, \) | \(\, \ds = \, \) | \(\ds 4 \times 48\) | $\sigma_1$ of $105$ |
$\blacksquare$
140
| \(\ds \map {\sigma_0} {140}\) | \(=\) | \(\, \ds 12 \, \) | \(\ds \) | $\sigma_0$ of $140$ | ||||||||||
| \(\ds \map \phi {140}\) | \(=\) | \(\, \ds 48 \, \) | \(\, \ds = \, \) | \(\ds 4 \times 12\) | $\phi$ of $140$ | |||||||||
| \(\ds \map {\sigma_1} {140}\) | \(=\) | \(\, \ds 336 \, \) | \(\, \ds = \, \) | \(\ds 7 \times 48\) | $\sigma_1$ of $140$ |
$\blacksquare$
168
| \(\ds \map {\sigma_0} {168}\) | \(=\) | \(\, \ds 16 \, \) | \(\ds \) | $\sigma_0$ of $168$ | ||||||||||
| \(\ds \map \phi {168}\) | \(=\) | \(\, \ds 48 \, \) | \(\, \ds = \, \) | \(\ds 3 \times 16\) | $\phi$ of $168$ | |||||||||
| \(\ds \map {\sigma_1} {168}\) | \(=\) | \(\, \ds 480 \, \) | \(\, \ds = \, \) | \(\ds 10 \times 48\) | $\sigma_1$ of $168$ |
$\blacksquare$
190
| \(\ds \map {\sigma_0} {190}\) | \(=\) | \(\, \ds 8 \, \) | \(\ds \) | $\sigma_0$ of $190$ | ||||||||||
| \(\ds \map \phi {190}\) | \(=\) | \(\, \ds 72 \, \) | \(\, \ds = \, \) | \(\ds 9 \times 8\) | $\phi$ of $190$ | |||||||||
| \(\ds \map {\sigma_1} {190}\) | \(=\) | \(\, \ds 360 \, \) | \(\, \ds = \, \) | \(\ds 5 \times 72\) | $\sigma_1$ of $190$ |
$\blacksquare$
210
| \(\ds \map {\sigma_0} {210}\) | \(=\) | \(\, \ds 16 \, \) | \(\ds \) | $\sigma_0$ of $210$ | ||||||||||
| \(\ds \map \phi {210}\) | \(=\) | \(\, \ds 48 \, \) | \(\, \ds = \, \) | \(\ds 3 \times 16\) | $\phi$ of $210$ | |||||||||
| \(\ds \map {\sigma_1} {210}\) | \(=\) | \(\, \ds 576 \, \) | \(\, \ds = \, \) | \(\ds 12 \times 48\) | $\sigma_1$ of $210$ |
$\blacksquare$
248
| \(\ds \map {\sigma_0} {248}\) | \(=\) | \(\, \ds 8 \, \) | \(\ds \) | $\sigma_0$ of $248$ | ||||||||||
| \(\ds \map \phi {248}\) | \(=\) | \(\, \ds 120 \, \) | \(\, \ds = \, \) | \(\ds 15 \times 8\) | $\phi$ of $248$ | |||||||||
| \(\ds \map {\sigma_1} {248}\) | \(=\) | \(\, \ds 480 \, \) | \(\, \ds = \, \) | \(\ds 4 \times 120\) | $\sigma_1$ of $248$ |
$\blacksquare$
264
| \(\ds \map {\sigma_0} {264}\) | \(=\) | \(\, \ds 16 \, \) | \(\ds \) | $\sigma_0$ of $264$ | ||||||||||
| \(\ds \map \phi {264}\) | \(=\) | \(\, \ds 80 \, \) | \(\, \ds = \, \) | \(\ds 5 \times 16\) | $\phi$ of $264$ | |||||||||
| \(\ds \map {\sigma_1} {264}\) | \(=\) | \(\, \ds 720 \, \) | \(\, \ds = \, \) | \(\ds 9 \times 80\) | $\sigma_1$ of $264$ |
$\blacksquare$
357
| \(\ds \map {\sigma_0} {357}\) | \(=\) | \(\, \ds 8 \, \) | \(\ds \) | $\sigma_0$ of $357$ | ||||||||||
| \(\ds \map \phi {357}\) | \(=\) | \(\, \ds 192 \, \) | \(\, \ds = \, \) | \(\ds 24 \times 8\) | $\phi$ of $357$ | |||||||||
| \(\ds \map {\sigma_1} {357}\) | \(=\) | \(\, \ds 576 \, \) | \(\, \ds = \, \) | \(\ds 3 \times 192\) | $\sigma_1$ of $357$ |
$\blacksquare$
420
| \(\ds \map {\sigma_0} {420}\) | \(=\) | \(\, \ds 24 \, \) | \(\ds \) | $\sigma_0$ of $420$ | ||||||||||
| \(\ds \map \phi {420}\) | \(=\) | \(\, \ds 96 \, \) | \(\, \ds = \, \) | \(\ds 4 \times 24\) | $\phi$ of $420$ | |||||||||
| \(\ds \map {\sigma_1} {420}\) | \(=\) | \(\, \ds 1344 \, \) | \(\, \ds = \, \) | \(\ds 14 \times 96\) | $\sigma_1$ of $420$ |
$\blacksquare$
570
| \(\ds \map {\sigma_0} {570}\) | \(=\) | \(\, \ds 16 \, \) | \(\ds \) | $\sigma_0$ of $570$ | ||||||||||
| \(\ds \map \phi {570}\) | \(=\) | \(\, \ds 144 \, \) | \(\, \ds = \, \) | \(\ds 9 \times 16\) | $\phi$ of $570$ | |||||||||
| \(\ds \map {\sigma_1} {570}\) | \(=\) | \(\, \ds 1440 \, \) | \(\, \ds = \, \) | \(\ds 10 \times 144\) | $\sigma_1$ of $570$ |
$\blacksquare$
616
| \(\ds \map {\sigma_0} {616}\) | \(=\) | \(\, \ds 16 \, \) | \(\ds \) | $\sigma_0$ of $616$ | ||||||||||
| \(\ds \map \phi {616}\) | \(=\) | \(\, \ds 240 \, \) | \(\, \ds = \, \) | \(\ds 15 \times 16\) | $\phi$ of $616$ | |||||||||
| \(\ds \map {\sigma_1} {616}\) | \(=\) | \(\, \ds 1440 \, \) | \(\, \ds = \, \) | \(\ds 6 \times 240\) | $\sigma_1$ of $616$ |
$\blacksquare$
630
| \(\ds \map {\sigma_0} {630}\) | \(=\) | \(\, \ds 24 \, \) | \(\ds \) | $\sigma_0$ of $630$ | ||||||||||
| \(\ds \map \phi {630}\) | \(=\) | \(\, \ds 144 \, \) | \(\, \ds = \, \) | \(\ds 6 \times 24\) | $\phi$ of $630$ | |||||||||
| \(\ds \map {\sigma_1} {630}\) | \(=\) | \(\, \ds 1872 \, \) | \(\, \ds = \, \) | \(\ds 13 \times 144\) | $\sigma_1$ of $630$ |
$\blacksquare$
Sources
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $210$