Divisor Function Calculator

In number theory, the divisor function σₓ(n) is the sum of the x^th powers of the divisors of n, that is

σₓ(n) = Σ d^x,

where the d ranges over the factors of n, including 1 and n. If x = 0, the function simply counts the number of factors. Sometimes σ₀(n) is denoted by d(n) or τ(n). When x = 1, the subscript 1 is often dropped.

Formula for σₓ(n)

For small integers, it is possible to enumerate the factors, but for larger numbers, it is more practical to use an explicit formula for σₓ(n) that gives the value of the function in terms of the prime factors of n.

First express n as the product of its prime factors:

n = p₁^a1⋅p₂^a2⋅...⋅p_k^ak

For values of x greater than 0, σₓ(n) is given by

σₓ(n) = Π[(p_i^(ai+1)x - 1)/(p_i^x - 1)]

If x = 0, the formula is

σ₀(n) = d(n) = Π(a_i + 1)

Example: To find the sum of the squared factors of 240, first decompose 240 into a product of powers of primes:

240 = 2⁴⋅3¹⋅5¹

Next, since x = 2, compute the product

[(2⁽⁴⁺¹⁾² - 1)/(2² - 1)]*[(3⁽¹⁺¹⁾² - 1)/(3² - 1)]*[(5⁽¹⁺¹⁾² - 1)/(5² - 1)]

= [1023/3]*[80/8]*[624/24]

= 341*10*26

= 88660

Properties and Identities

The divisor function is multiplicative, meaning that if m and n are relatively prime then

    σₓ(m)σₓ(n) = σₓ(mn).

By the Möbius Inversion Formula,

    n^x = Σ μ(d)σₓ(n/d)

where μ(n) is the Möbius Mu function and d ranges over the factors of n. Another relation is

    Σσₓ(n)/n^s = ζ(s)ζ(s-x),

where n ranges from 1 to infinity and where ζ(s) is the Riemann Zeta function defined by

    ζ(s) = Σ1/n^s

from n = 1 to infinity. A summation identity involving the floor function and divisor function is

    Σ σₓ(k) = Σ k^x⌊n/k⌋

where k ranges from 1 to n. Some identities that involve both the divisor function and Euler's Totient function are

    Σ σ₁(d)φ(n/d) = nσ₀(n) and

    Σ σ₀(d)φ(n/d) = σ₁(n)

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