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NumberTheory

  

SumOfDivisors

  

sum of powers of the divisors

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

SumOfDivisors(n)

 

SumOfDivisors(n, k)

 

sigma(n)

σn

sigma[k](n)

σkn

tau(n)

τn

Parameters

n

-

integer

k

-

(optional) non-negative integer; defaults to 1

Description

• 

The SumOfDivisors(n) command computes the sum of the positive divisors of n.

• 

If n has divisors di for i from 1 to r, then SumOfDivisors(n, k) computes the sum of the powers of the positive divisors and is equal to i=1rdik.

• 

sigma (σ) is an alternate calling sequence for SumOfDivisors, where sigma[k](n) is equal to SumOfDivisors(n, k) and k defaults to 1 if the index is omitted.

• 

tau (τ) counts the number of divisors of n. That is, tau(n) is equal to SumOfDivisors(n, 0).

• 

Every prime number divides 0 evenly, so 0 has infinitely many prime factors. For consistency with, for example, the Divisors command, SumOfDivisors(0) returns an error, as does SumOfDivisors(0, k) for any k.

• 

You can enter the commands sigma and tau using either the 1-D or 2-D calling sequence. For example, sigma(8) is equivalent to σ8, sigma[2](8) is equivalent to σ28, and tau(8) is equivalent to τ8.

• 

If i=1mpiai is the prime factorization of the n, then SumOfDivisors is given by the formula i=1mpiai+1k1pik1 if k is nonzero and by the formula i=kmai+1 if k is zero.

Examples

withNumberTheory:

Divisors12

1,2,3,4,6,12

(1)

SumOfDivisors12

28

(2)

τ12

6

(3)

Divisors52

1,2,4,13,26,52

(4)

σ252

3570

(5)

SumOfDivisors52,2

3570

(6)

SumOfDivisors0

Error, (in NumberTheory:-SumOfDivisors) 0 has infinitely many prime factors

τ0

Error, (in NumberTheory:-SumOfDivisors) 0 has infinitely many prime factors

Compatibility

• 

The NumberTheory[SumOfDivisors] command was introduced in Maple 2016.

• 

For more information on Maple 2016 changes, see Updates in Maple 2016.

See Also

NumberTheory

NumberTheory[Divisors]