PlanetMath: $\sigma$ function

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$\sigma$ function

(Definition)

Given a positive integer , the sum of the integers $0 < d \le n$ such that $d\vert n$ is the value of the sum of divisors function for , often symbolized by a lowercase sigma. Thus,

$\displaystyle \sigma(n) = \sum_{d\vert n} d.$

Sometimes this function is referred to as $\sigma_1(n)$ , highlighting its relation to the divisor function.

Given coprime integers and (that is, $\gcd(m, n) = 1$ ) then $\sigma(mn) = \sigma(m)\sigma(n)$ , meaning that the sum of divisors function is a multiplicative function.

" $\sigma$ function" is owned by CompositeFan.

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Other names:

divisor sigma, sum of divisors function, $\sigma_1$ function

Attachments:: iterated sum of divisors function (Definition) by CompositeFan

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Cross-references: multiplicative function, coprime, divisor function, relation, function, sum, integer, positive
There are 13 references to this entry.

This is version 3 of $\sigma$ function, born on 2006-07-28, modified 2006-08-03.
Object id is 8188, canonical name is SumOfDivisorsFunction.
Accessed 2227 times total.

Classification:

AMS MSC:

11A25 (Number theory :: Elementary number theory :: Arithmetic functions; related numbers; inversion formulas)

Pending Errata and Addenda

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