Number of divisors function

This article needs more work.

Please help by expanding it!

The divisor function 

σk (n)

for a positive integer 

n

is defined as the sum of the 

k

th powers of the divisors of 

n

$\sigma _{k}(n):=\sum _{d|n}d^{k}.\,$

For 

k = 0

we get

$\sigma _{0}(n):=\sum _{d|n}d^{\,0}=\sum _{d|n}1=:D(n)=:d(n)=:\nu (n)=:\tau (n),\,$

where 

τ (n)

is the number of divisors function. The notations 

d (n)

, 

ν (n)

, and 

τ (n)

are sometimes used for 

σ0(n)

, which gives the number of divisors of 

n

. For 

n   ≥   1

, the number of divisors is the number of restricted partitions with parts of equal size.

A000005 

d (n)

(also called 

τ (n)

or 

σ0(n)

), the number of divisors of 

n, n   ≥   1

.

 

{1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 6, 2, 4, 4, 5, 2, 6, 2, 6, 4, 4, 2, 8, 3, 4, 4, 6, 2, 8, 2, 6, 4, 4, 4, 9, 2, 4, 4, 8, 2, 8, 2, 6, 6, 4, 2, 10, 3, 6, 4, 6, 2, 8, 4, 8, 4, 4, 2, 12, 2, 4, 6, 7, 4, 8, 2, 6, 4, 8, 2, ...}

Number of divisors greater than 1, number of divisors smaller than n

The number of divisors greater than 1 (same as number of aliquot divisors, i.e. number of divisors smaller than n) of 

n

is one less than the number of divisors of 

n

.

Formulae for the number of divisors function

From the prime factorization of 

n

$n=\prod _{i=1 \atop {{p_{i}}^{\alpha _{i}}\parallel n,\,\alpha _{i}\geq 1}}^{\omega (n)}{p_{i}}^{\alpha _{i}},\,$

where the 

pi

are the distinct prime factors of 

n

and 

ω (n)

is the number of distinct prime factors of 

n

, we obtain the number of divisors of 

n

$\sigma _{0}(n)=\prod _{i=1}^{\omega (n)}(1+\alpha _{i}),\,$

since for each 

pi

we can choose the exponent from 

0

to 

αi

to build a divisor of 

n

.

Generating function of number of divisors function

The generating function is

$G_{\{\sigma _{0}(n)\}}(x):=\sum _{n=1}^{\infty }\sigma _{0}(n)\,x^{n}=\sum _{n=0}^{\infty }{\frac {x^{n}}{1-x^{n}}}.\,$

This is usually called THE Lambert series (see Knopp, Titchmarsh).

Dirichlet generating function of number of divisors function

The Dirichlet generating function is

$D_{\{\sigma _{0}(n)\}}(s):=\sum _{n=1}^{\infty }{\frac {\sigma _{0}(n)}{n^{s}}}=(\zeta (s))^{2}.\,$

Number of ways of factoring n with all factors greater than 1

What is the relation between the number of ways of factoring n with all factors greater than 1 and the number of divisors greater than 1 (same as number of aliquot divisors) of 

n

?

Number of even divisors

(...)

Number of odd divisors

(...)

Number of divisors of form 4m + 1

(...)

Number of divisors of form 4m + 3

(...)

(number of divisors of form 4m + 1) − (number of divisors of form 4m + 3)

A002654 Number of ways of writing 

n

as a sum of at most two nonzero squares, where order matters; also (number of divisors of 

n

of form 

4 m + 1

) 

−

(number of divisors of 

n

of form 

4 m + 3

).

 

{1, 1, 0, 1, 2, 0, 0, 1, 1, 2, 0, 0, 2, 0, 0, 1, 2, 1, 0, 2, 0, 0, 0, 0, 3, 2, 0, 0, 2, 0, 0, 1, 0, 2, 0, 1, 2, 0, 0, 2, 2, 0, 0, 0, 2, 0, 0, 0, 1, 3, 0, 2, 2, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 1, 4, 0, 0, 2, 0, 0, 0, 1, ...}

A213408 Sequence A002654 with the zero terms omitted.

 

{1, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 2, 3, 2, 2, 1, 2, 1, 2, 2, 2, 2, 1, 3, 2, 2, 2, 2, 1, 4, 2, 1, 2, 2, 2, 1, 2, 4, 2, 2, 2, 1, 3, 2, 2, 2, 2, 2, 2, 2, 1, 2, 4, 1, 4, 2, 2, 1, 4, 2, 2, 2, 2, 2, 2, 1, 2, 3, 4, 2, 2, 2, ...}

See also

• Numbers with relatively many divisors

• Divisor function
  • Number of divisors function (number of divisors)
    • Number of even divisors function (number of even divisors)
  • Sum of divisors function (sum of divisors)
    • Sum of even divisors function (sum of even divisors)
  • Divisorial function (divisorial, product of divisors)
    • Product of even divisors function (product of even divisors)

• Odd divisor function
  • Number of odd divisors function (number of odd divisors)
    • Number of divisors of form 4m + 1
    • Number of divisors of form 4m + 3
    • (number of divisors of form 4m + 1) − (number of divisors of form 4m + 3)
  • Sum of odd divisors function (sum of odd divisors)
    • Sum of divisors of form 4m + 1
    • Sum of divisors of form 4m + 3
    • (sum of divisors of form 4m + 1) − (sum of divisors of form 4m + 3)
  • Product of odd divisors function (product of odd divisors)
    • Product of divisors of form 4m + 1
    • Product of divisors of form 4m + 3
    • (product of divisors of form 4m + 1) − (product of divisors of form 4m + 3)

Arithmetic function templates

• {{divisor function}} or {{sigma k}} arithmetic function template (for 

  k   ≥   0

  )
• {{number of divisors}} or {{sigma 0}} or {{tau}} arithmetic function template (for 

  k = 0

  )
• {{sum of divisors}} or {{sigma 1}} or {{sigma}} arithmetic function template (for 

  k = 1

  )

References

• Burton, D. M. (1989). Elementary Number Theory (4th ed.). Boston, MA: Allyn and Bacon. 

• Hardy, G. H.; Wright, E. M. (1979). An Introduction to the Theory of Numbers (5th ed.). Oxford, England: Oxford University Press. pp. 354-355. 

• Ore, Ø. (1988). Number Theory and Its History. New York: Dover.