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Number of divisors function
From OeisWiki
σk (n) 
n 
k 
n 
k = 0 
τ (n) 
d (n) 
ν (n) 
τ (n) 
σ0(n) 
n 
n ≥ 1 
A000005
d (n) 
τ (n) 
σ0(n) 
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, ...}
Contents
 1 Number of divisors greater than 1, number of divisors smaller than n
 2 Formulae for the number of divisors function
 3 Generating function of number of divisors function
 4 Dirichlet generating function of number of divisors function
 5 Number of ways of factoring n with all factors greater than 1
 6 Number of even divisors
 7 Number of odd divisors
 8 See also
 9 Notes
 10 References
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) ofn 
n 
Formulae for the number of divisors function
From the prime factorization ofn 
pi 
n 
ω (n) 
n 
n 
pi 
0 
αi 
n 
Generating function of number of divisors function
The generating function is
This is usually called THE Lambert series (see Knopp, Titchmarsh).
Dirichlet generating function of number of divisors function
The Dirichlet generating function is
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) ofn 
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 writingn 
n 
4 m + 1 
− 
n 
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
 Divisor function
 Odd divisor function
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
Notes
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. 354355.
 Ore, Ø. (1988). Number Theory and Its History. New York: Dover.