NumberTheory
SumOfDivisors
sum of powers of the divisors
Calling Sequence
Parameters
Description
Examples
Compatibility
SumOfDivisors(n)
SumOfDivisors(n, k)
sigma(n)
σn
sigma[k](n)
σkn
tau(n)
τn
n
-
integer
k
(optional) non-negative integer; defaults to 1
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+1k−1pik−1 if k is nonzero and by the formula ∏i=1mai+1 if k is zero.
withNumberTheory:
Divisors12
1,2,3,4,6,12
SumOfDivisors12
28
τ12
6
Divisors52
1,2,4,13,26,52
σ252
3570
SumOfDivisors52,2
SumOfDivisors0
Error, (in NumberTheory:-SumOfDivisors) 0 has infinitely many prime factors
τ0
The NumberTheory[SumOfDivisors] command was introduced in Maple 2016.
For more information on Maple 2016 changes, see Updates in Maple 2016.
See Also
NumberTheory[Divisors]
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