Sum of Divisors

Inputs

Integer

n

A positive integer, arbitrarily large.

Generalized divisor sum

\sigma_k(n)

Exponent

k

for

\sigma_k(n)

Integer k ≥ 0.

$\sigma_k(n)=\sum_{d\mid n}d^{k}$ over all divisors $d$ of $n$ ; $\sigma_0(n)=\tau(n)$ and $\sigma_1(n)=\sigma(n)$ .

Factorization difficulty depends on the size of the factors, not just the size of the input. The calculator may return a partial factorization for difficult inputs. Computation runs locally in your browser.

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Result

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Prime factorization

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How it works

$\sigma$ is multiplicative, and for a prime power the divisors are $1,p,\dots,p^{a}$ , so

\sigma(p^{a})=1+p+\cdots+p^{a}=\frac{p^{a+1}-1}{p-1}.

Therefore

\sigma(n)=\prod_i\left(1+p_i+\cdots+p_i^{a_i}\right).

More generally,

\sigma_k(n)=\sum_{d\mid n}d^{k}=\prod_i\left(1+p_i^{k}+p_i^{2k}+\cdots+p_i^{a_ik}\right),

so $\sigma_0(n)=\tau(n)$ and $\sigma_1(n)=\sigma(n)$ . Changing only $k$ does not refactor $n$ . $\sigma(1)=\sigma_k(1)=1$ .

Inputs

Result

Prime factorization

$\sigma(n)$ — Sum of Divisors

$\sigma_k(n)$ — Generalized divisor sum

How it works

Sum of Divisors

Inputs

Result

Prime factorization

σ(n) — Sum of Divisors

σk(n) — Generalized divisor sum

How it works

$\sigma(n)$ — Sum of Divisors

$\sigma_k(n)$ — Generalized divisor sum