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# Number of divisors function

(Redirected from Number of divisors of n)

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
${\displaystyle \sigma _{k}(n):=\sum _{d|n}d^{k}.\,}$
For
 k = 0
we get
${\displaystyle \sigma _{0}(n):=\sum _{d|n}d^{\,0}=\sum _{d|n}1=:D(n)=:d(n)=:\nu (n)=:\tau (n),\,}$
where
 τ (n)
is the number of divisors function. The notations
 d (n)
[1],
 ν (n)
[2], and
 τ (n)
[3] 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
${\displaystyle n=\prod _{i=1 \atop {{p_{i}}^{\alpha _{i}}\parallel n,\,\alpha _{i}\geq 1}}^{\omega (n)}{p_{i}}^{\alpha _{i}},\,}$
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
${\displaystyle \sigma _{0}(n)=\prod _{i=1}^{\omega (n)}(1+\alpha _{i}),\,}$
since for each
 pi
we can choose the exponent from
 0
to
 αi
to build a divisor of
 n
.[4]

### Generating function of number of divisors function

The generating function is

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

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

### Dirichlet generating function of number of divisors function

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

### 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 odd divisors

(...)

(...)

(...)

### (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, ...}

## Notes

1. Hardy and Wright 1979, p. 239.
2. Ore 1988, p. 86.
3. Burton 1989, p. 128.
4. Charles Vanden Eynden, Elementary Number Theory, 2nd Edition. Long Grove, Illinois: Waveland Press (2001): 71

## 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.