Numbers such that Divisor Count divides Phi divides Divisor Sum/Mistake

Source Work

1997: David Wells: Curious and Interesting Numbers (2nd ed.):

The Dictionary

$210$

Mistake

$\map \phi {210} = 48$ is a factor of $\map \sigma {210} = 576$, and $\map d {210} = 16$ divides both. The sequence of numbers with both these properties starts: $1 \quad 3 \quad 15 \quad 30 \quad 35 \quad 52 \quad 70 \quad 78 \quad 105 \quad 140 \quad 168 \quad 190 \quad 210 \ldots$

The list is incorrect.

$52$ does not have this property:

\(\ds \map {\sigma_0} { 52 }\)

\(=\)

\(\ds 6\)

$\sigma_0$ of $52$

\(\ds \map \phi { 52 }\)

\(=\)

\(\ds 24\)

$\phi$ of $52$

\(\ds \map {\sigma_1} { 52 }\)

\(=\)

\(\ds 98\)

$\sigma_1$ of $52$

and neither $6$ nor $24$ divide $98 = 2 \times 7^2$.

Instead it appears that $56$ is the correct number here:

\(\ds \map {\sigma_0} { 56 }\)

\(=\)

\(\ds 8\)

$\sigma_0$ of $56$

\(\ds \map \phi { 56 }\)

\(=\)

\(\ds 24\)

$\phi$ of $56$

\(\ds \map {\sigma_1} { 56 }\)

\(=\)

\(\ds 120\)

$\sigma_1$ of $56$

The list should be:

$1, 3, 15, 30, 35, 56, 70, 78, 105, 140, 168, 190, 210$

Sources

• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $210$