Divisor Function


The divisor function sigma_k(n) for n an integer is defined as the sum of the kth powers of the (positive integer) divisors of n,


It is implemented in the Wolfram Language as DivisorSigma[k, n].

The notations d(n) (Hardy and Wright 1979, p. 239), nu(n) (Ore 1988, p. 86), and tau(n) (Burton 1989, p. 128) are sometimes used for sigma_0(n), which gives the number of divisors of n. Rather surprisingly, the number of factors of the polynomial a^n-b^n are also given by d(n). The values of sigma_0(n) can be found as the inverse Möbius transform of 1, 1, 1, ... (Sloane and Plouffe 1995, p. 22). Heath-Brown (1984) proved that sigma_0(n)=sigma_0(n+1) infinitely often. The numbers having the incrementally largest number of divisors are called highly composite numbers. The function sigma_0(n) satisfies the identities


where the p_i are distinct primes and p_1^(a_1)p_2^(a_2)... is the prime factorization of a number n.

The divisor function sigma_0(n) is odd iff n is a square number.

The function sigma_1(n) that gives the sum of the divisors of n is commonly written without the subscript, i.e., sigma(n).

As an illustrative example of computing sigma_k(n), consider the number 140, which has divisors d_i=1, 2, 4, 5, 7, 10, 14, 20, 28, 35, 70, and 140, for a total of N=12 divisors in all. Therefore,


The following table summarized the first few values of sigma_k(n) for small k and n=1, 2, ....

kOEISsigma_k(n) for n=1, 2, ...
0A0000051, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 6, ...
1A0002031, 3, 4, 7, 6, 12, 8, 15, 13, 18, ...
2A0011571, 5, 10, 21, 26, 50, 50, 85, 91, 130, ...
3A0011581, 9, 28, 73, 126, 252, 344, 585, 757, 1134, ...

The sum of the divisors of n excluding n itself (i.e., the proper divisors of n) is called the restricted divisor function and is denoted s(n). The first few values are 0, 1, 1, 3, 1, 6, 1, 7, 4, 8, 1, 16, ... (OEIS A001065).

The sum of divisors sigma_1(N) can be found as follows. Let N=ab with a!=b and (a,b)=1. For any divisor d of N, d=a_ib_i, where a_i is a divisor of a and b_i is a divisor of b. The divisors of a are 1, a_1, a_2, ..., and a. The divisors of b are 1, b_1, b_2, ..., b. The sums of the divisors are then


For a given a_i,


Summing over all a_i,


so sigma_1(N)=sigma_1(ab)=sigma_1(a)sigma_1(b). Splitting a and b into prime factors,


For a prime power p_i^(alpha_i), the divisors are 1, p_i, p_i^2, ..., p_i^(alpha_i), so


For N, therefore,


(Berndt 1985).

For the special case of N a prime, (14) simplifies to


Similarly, for N a power of two, (14) simplifies to


The identities (◇) and (◇) can be generalized to


Sums involving the divisor function are given by


for s>1,


for s>2, and more generally,


for s>1 and k>=0 (Hardy and Wright 1979, p. 250).

A generating function for sigma_0(n) is given by the Lambert series


where phi_q(x) is a q-polygamma function.

The sigma_1(n) function has the series expansion


(Hardy 1999). Ramanujan gave the beautiful formula


where zeta(n) is the zeta function and R[s],R[s-a],R[s-b],R[s-a-b]>1 (Wilson 1923), which was used by Ingham in a proof of the prime number theorem (Hardy 1999, pp. 59-60). This gives the special case


(Hardy 1999, p. 59).

Gronwall's theorem states that


where gamma is the Euler-Mascheroni constant (Hardy and Wright 1979, p. 266; Robin 1984). This can be written as an explicit inequality as


where gamma is the Euler-Mascheroni constant and where equality holds for n=12, giving


(Robin 1984, Theorem 2). In fact, the constant term can be dropped if the Riemann hypothesis holds, since the Riemann hypothesis is equivalent to the statement that


for all n>=5041 (Robin 1984, Theorem 1).

sigma_1(n) is a power of 2 iff n=1 or n is a product of distinct Mersenne primes (Sierpiński 1958/59, Sivaramakrishnan 1989, Kaplansky 1999). The first few such n are 1, 3, 7, 21, 31, 93, 127, 217, 381, 651, 889, 2667, ... (OEIS A046528), and the powers of 2 these correspond to are 0, 2, 3, 5, 5, 7, 7, 8, 9, 10, 10, 12, 12, 13, 14, ... (OEIS A048947).

Curious identities derived using modular form theory are given by


(Apostol 1997, p. 140), together with


(M. Trott, pers. comm.).

The divisor function sigma_1(n) (and, in fact, sigma_k(n) for k>=1) is odd iff n is a square number or twice a square number. The divisor function sigma_1(n) satisfies the congruence

 nsigma_1(n)=2 (mod phi(n)),

for all primes and no composite numbers with the exception of 4, 6, and 22 (Subbarao 1974).

The number of divisors d(n) is prime whenever sigma_1(n) itself is prime (Honsberger 1991). Factorizations of sigma_1(p^a) for prime p are given by Sorli.


In 1838, Dirichlet showed that the average number of divisors of all numbers from 1 to n is asymptotic to


(Conway and Guy 1996; Hardy 1999, p. 55; Havil 2003, pp. 112-113), as illustrated above, where the thin solid curve plots the actual values and the thick dashed curve plots the asymptotic function. This is related to the Dirichlet divisor problem, which seeks to find the "best" coefficient theta in


(Hardy and Wright 1979, p. 264).

The summatory functions for sigma_a with a>1 are


For a=1,


(Hardy and Wright 1979, p. 266).

The divisor function can also be generalized to Gaussian integers. The definition requires some care since in principle, there is ambiguity as to which of the four associates is chosen for each divisor. Spira (1961) defines the sum of divisors of a complex number z by factoring z into a product of powers of distinct Gaussian primes,


where epsilon is a unit and each p_i lies in the first quadrant of the complex plane, and then writing


This makes sigma a multiplicative function and also gives |sigma_1(z)|>=z. This extension is implemented in the Wolfram Language as DivisorSigma[1, z, GaussianIntegers -> True]. The following table gives sigma_1(a+ib) for small nonnegative values of a and b.


See also

Dirichlet Divisor Problem, Distinct Prime Factors, Divisor, Divisor Product, Even Divisor Function, Factor, Fermat's Divisor Problem, Greatest Prime Factor, Gronwall's Theorem, Highly Composite Number, Least Prime Factor, Multiperfect Number, Odd Divisor Function, Ore's Conjecture, Perfect Number, Prime Factor, Refactorable Number, Restricted Divisor Function, Robin's Theorem, Silverman Constant, Sociable Numbers, Sum of Squares Function, Superabundant Number, Tau Function, Totient Function, Totient Valence Function, Twin Peaks, Unitary Divisor Function Explore this topic in the MathWorld classroom

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Divisor Function

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Weisstein, Eric W. "Divisor Function." From MathWorld--A Wolfram Web Resource.

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