Sum of divisors

2020 Mathematics Subject Classification: Primary: 11A25 Secondary: 11A51 [MSN][ZBL]

of a natural number $n$

The sum of the positive integers divisors of a natural number $n$, including $1$ and $n$: $$\sigma (n)=\sum _{d|n}d.$$ More generally, the function ${\sigma }_k$ is defined as $${\sigma }_k(n)=\sum _{d|n}{d}^k.$$ so that $\sigma ={\sigma }_1$ and the number of divisors function $\tau ={\sigma }_0$.

These are multiplicative arithmetic functions with Dirichlet series $$\sum _{n=1}^{\mathrm{\infty }}{\sigma }_k(n)n^{-s}=\prod _p{\left((1-p^{-s})(1-p^{k-s})\right)}^{-1}=\zeta (s)\zeta (s-k).$$

The average order of $\sigma (n)$ is given by $$\sum _{n\le x}\sigma (n)=\frac{{\pi }^2}{12}{x}^2+O(x\log x).$$

There are a number of well-known classes of number characterised by their divisor sums.

A perfect number $n$ is the sum of its aliquot divisors (those divisors other than $n$ itself), so $\sigma (n)=2n$. The even perfect numbers are characterised in terms of Mersenne primes $P=2^p-1$ as $n=2^{p-1}.P$: it is not known if there are any odd perfect numbers. An almost perfect number $n$ similarly has the property that $\sigma (n)=2n-1$: these include the powers of 2. A quasiperfect number is defined by $\sigma (n)=2n+1$: it is not known if any exists. See also Descartes number.

References[edit]

• Kishore, Masao. "On odd perfect, quasiperfect, and odd almost perfect numbers". Mathematics of Computation 36 (1981) 583–586. Zbl 0472.10007
• G. Tenenbaum, Introduction to Analytic and Probabilistic Number Theory, Cambridge Studies in Advanced Mathematics 46, Cambridge University Press (1995) ISBN 0-521-41261-7 Zbl 0831.11001