Counterbalanced Numbers

 

Today's daily count turned up counterbalanced numbers that apparently have the property that:$$ \frac{\phi(n)}{\sigma(n)-n}$$is an integer. Of course, this prompted me to revisit the phi and sigma functions because I'm regularly forgetting what they signify. To recapitulate, the phi function is the Euler totient function defined by WolframAlpha as:

The totient function \(\phi(n) \), also called Euler's totient function, is defined as the number of positive integers \(\leq n\) that are relatively prime to (i.e., do not contain any factor in common with) \(n\), where 1 is counted as being relatively prime to all numbers. Since a number less than or equal to and relatively prime to a given number is called a totative, the totient function \(\phi(n)\) can be simply defined as the number of totatives of \(n\). For example, there are eight totatives of 24 (1, 5, 7, 11, 13, 17, 19, and 23), so \( \phi(24)=8\).

In this case, \( \phi(24307)=23976\). The sigma function gives the number of divisors of a number and in this case \( \sigma(24307)=24640\). Thus:$$ \sigma(24307)-24307=24640-24307=333 \text{ and }\\ \frac{\phi(24307)}{333}=\frac{23976}{333}=72$$Actually the sigma function is a little more complicated because technically is comes with a subscript that can have any integer value, positive or negative. Let's say the subscript is \(k\), then the sigma function represents the sum of the \(k^{th}\) powers of the divisors. Thus when \(k=0\), it returns the number of divisors because the \(0^{th}\) powers of the divisors are 1. When \(k=1\), it returns the sum of the divisors and so it really is the \(k=1\) case that the sigma function is being used above.

on July 5th 2021