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NumberTheory

  

SumOfDivisors

  

sum of powers of the divisors

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

SumOfDivisors(n, k)

sigma[k](n)

tau(n)

Parameters

n

-

non-zero integer

k

-

(optional) non-negative integer; defaults to 1

Description

• 

The SumOfDivisors command computes the sum of powers of the positive divisors of n.

• 

If n has divisors di for i from 1 to r, then SumOfDivisors(n, k) 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, i.e. tau(n) is equal to SumOfDivisors(n, 0).

• 

If i=1mpiai is the prime factorization of the n, then SumOfDivisors is given by the formula i=1mpiai+1k1pik1 if k is non-zero 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)

σ[2]52

3570

(5)

SumOfDivisors52,2

3570

(6)

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