\[\sigma _{k}(n)\]
The divisor function is defined as the sum of k th powers of the positive integer divisors of n.
\[\sigma _{k}(n)\equiv\sum_{d|n}^{ } d^{k}\]
\[\sigma _{1}(n)\]
Sum of divisors of n.
A000203 Sum of the divisors of n.
A000203(n) = A001065(n) + n .
\[\tau (n)=\sum_{d|n}1\]
Add 1 every time we encounter a divider $\leqslant$ n .
sum([1 for d in divisors(n)])
\[\sigma _{0}(n)\]
\[\tau (n)\]
\[d(n)\]
Number of divisors of n.
A000005 Number of divisors of n.
If A000005(n) is odd, n is a square (a power of 2).
A000290 n^2
\[s_{1}(n) \equiv \sigma_{1}(n)-n\]
Sum of proper divisors of n.
A001065 Sum of proper divisors.
This is the Restricted Divisor Function.
A001065(n) = A000203(n) - n .
\[s_{0}(n) \equiv \sigma_{0}(n)-1\]
Number of proper divisors of n.
A032741 Number of proper divisors of n.
This is the Proper Divisors Function.
A032741(n) = A000005(n) - 1 .
A032741(n) = A002541(n+1) / A002541(n) .
A002541 Sum_{k=1..n-1} floor((n-k)/k).
\[\sigma _{1}(n)-\sigma _{0}(n)=A000203(n)-A000005(n)=A065608(n)\]
A065608 Sum of divisors of n minus the number of divisors of n.
\[\sigma _{0}(n)-2=A000005(n)-2=A070824(n)\]
A070824 Number of nontrivial divisors of n.
A027750 The divisors of n.
n=25
divisors(n)
[1, 5, 25]