Integers whose Divisor Sum equals Half Phi times Divisor Count
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Theorem
The following positive integers $n$ have the property where:
- $\map {\sigma_1} n = \dfrac {\map \phi n \times \map {\sigma_0} n} 2$
where:
- $\map {\sigma_1} n$ denotes the divisor sum function: the sum of the divisors of $n$
- $\map \phi n$ denotes the Euler $\phi$ function: the count of positive integers smaller than of $n$ which are coprime to $n$
- $\map {\sigma_0} n$ denotes the divisor count function: the count of the divisors of $n$:
These positive integers are:
- $35, 105, \ldots$
Proof
We have:
| \(\ds \map \phi {35}\) | \(=\) | \(\ds 24\) | $\phi$ of $35$ | |||||||||||
| \(\ds \map {\sigma_0} {35}\) | \(=\) | \(\ds 4\) | $\sigma_0$ of $35$ | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \map \phi {35} \times \map {\sigma_0} {35}\) | \(=\) | \(\ds \dfrac {24 \times 4} 2\) | |||||||||||
| \(\ds \) | \(=\) | \(\ds 48\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \map {\sigma_1} {35}\) | $\sigma_1$ of $35$ |
| \(\ds \map \phi {105}\) | \(=\) | \(\ds 48\) | $\phi$ of $105$ | |||||||||||
| \(\ds \map {\sigma_0} {105}\) | \(=\) | \(\ds 8\) | $\sigma_0$ of $105$ | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \map \phi {105} \times \map {\sigma_0} {105}\) | \(=\) | \(\ds \dfrac {48 \times 8} 2\) | |||||||||||
| \(\ds \) | \(=\) | \(\ds 192\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \map {\sigma_1} {105}\) | $\sigma_1$ of $105$ |
$\blacksquare$
Also see
Historical Note
The intent of this result is unclear. Its statement by David Wells in his Curious and Interesting Numbers, 2nd ed. of $1997$ was erroneous, but no indication was given as to where it originated.
The On-Line Encyclopedia of Integer Sequences suggests that this result may be intended as:
- $\map {\sigma_1} n = \map \phi n \times \map j n$
where $\map j n$ is the count of $d \divides n$ such that $d \ge 3$ and $1 \le \dfrac n d \le d - 2$.
In such a case, the sequence begins:
- $35, 105, 248, 418, 594, 744, 812, 1254, \ldots$
It is also possible that the result may also have been intended to be:
- $\map {\sigma_1} n = \map \phi n \times \map k n$
where $\map k n$ is the count of $d \divides n$ such that $d < \sqrt n$.
Sources
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $105$