Is there a recurrence relation that describes the aliquot sum?

Question

I am trying to understand the dynamics of the aliquot sum.

I am wondering if a recurrence relation exists?

For example, would this work:

• Let $s(x)$ be the aliquot sum for $x$
• Let $p$ be a prime
• If $x$ is $1$, then $s(x)=0$
• If $x$ is prime, then $s(x)=1$
• If $p \nmid x$, then $s(px) = s(x) + ps(x) + x$
• If $p | x$, then $s(px) = s(x) + x$

Thanks.

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Edit: Made updates based on comments received by Mason.

Answer 1

Define $f(x)+x=\sigma(x)=\sum_{d|x}d$. Then $f(x)$ is our aliquot function.

For coprime numbers $a,b$ Then $\sigma(a b)= \sigma(a)\sigma(b) $ so

$f(ab)+ab=(f(a)+a)(f(b)+b)$

This implies that

$f(ab)= f(a)f(b)+bf(a)+af(b)$ for coprime $a,b$.

While we're talking about recurrence we should mention this amazing recurrence formula from Euler that can be found here.