# Sum of divisors

2010 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 $$ \sum_{n=1}^\infty \sigma_k(n) n^{-s} = \prod_p \left({(1-p^{-s})(1-p^{k-s}) }\right)^{-1} = \zeta(s) \zeta(s-k)\ . $$

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

- Kishore, Masao. "On odd perfect, quasiperfect, and odd almost perfect numbers". Mathematics of Computation
**36**(1981) 583–586. ISSN 0025-5718. 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

**How to Cite This Entry:**

Sum of divisors.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Sum_of_divisors&oldid=37103