The document discusses divisor functions and partition functions in number theory. It defines the divisor function τ(n) as counting the number of divisors of an integer n. It provides examples of calculating τ(n) and describes properties such as τ(p)=2 for prime p. The partition function P(n) counts the number of ways to write n as a sum of positive integers, ignoring order. Recursion formulas and generating functions are presented for both functions.

1. DIVISOR FUNCTION

2. Divisor Function
• In mathematics, and specifically in number
theory, a divisor function is an arithmetic
function related to the divisors of an integer.
When referred to as the divisor function, it
counts the number of divisors of an integer.
• A related function is the divisor summatory
function, which, as the name implies, is a sum
over the divisor function.

3. Number of Divisors
Function Tau (τ) or d = number of divisors
function
τ(n) = number of divisors of n (including 1 & n)
Example:
d(12) = τ(12) 12 {1}

4. Number of Divisors
Function Tau (τ) or d = number of divisors
function
τ(n) = number of divisors of n (including 1 & n)
Example:
d(12) = τ(12) 12 {1,2}

5. Number of Divisors
Function Tau (τ) or d = number of divisors
function
τ(n) = number of divisors of n (including 1 & n)
Example:
d(12) = τ(12) 12 {1,2,3}

6. Number of Divisors
Function Tau (τ) or d = number of divisors
function
τ(n) = number of divisors of n (including 1 & n)
Example:
d(12) = τ(12) 12 {1,2,3,4}

7. Number of Divisors
Function Tau (τ) or d = number of divisors
function
τ(n) = number of divisors of n (including 1 & n)
Example:
d(12) = τ(12) 12 {1,2,3,4,6}

8. Number of Divisors
Function Tau (τ) or d = number of divisors
function
τ(n) = number of divisors of n (including 1 & n)
Example:
d(12) = τ(12) 12 = {1,2,3,4,6,12}

9. Number of Divisors
Function Tau (τ) or d = number of divisors
function
τ(n) = number of divisors of n (including 1 & n)
Example:
d(12) = τ(12)
τ(12) = 6
12 {1,2,3,4,6,12} = 6

10. Number of Divisors
Function Tau (τ) or d = number of divisors
function
τ(n) = number of divisors of n (including 1 & n)
Example:
τ(16) =

11. Number of Divisors
Function Tau (τ) or d = number of divisors
function
τ(n) = number of divisors of n (including 1 & n)
Example:
τ(16) = 1

12. Number of Divisors
Function Tau (τ) or d = number of divisors
function
τ(n) = number of divisors of n (including 1 & n)
Example:
τ(16) = 1,2

13. Number of Divisors
Function Tau (τ) or d = number of divisors
function
τ(n) = number of divisors of n (including 1 & n)
Example:
τ(16) = 1,2,4

14. Number of Divisors
Function Tau (τ) or d = number of divisors
function
τ(n) = number of divisors of n (including 1 & n)
Example:
τ(16) = 1,2,4,8

15. Number of Divisors
Function Tau (τ) or d = number of divisors
function
τ(n) = number of divisors of n (including 1 & n)
Example:
τ(16) = 1,2,4,8,16
τ(16) = 5

16. Number of Divisors
• If p is prime
• τ(2n)
τ(p) = 2

17. Number of Divisors
• If p is prime
• τ(2n) = 1
τ(p) = 2

18. Number of Divisors
• If p is prime τ(p) = 2
• τ(2n) = 1,2

19. Number of Divisors
• If p is prime τ(p) = 2
• τ(2n) = 1,2, 22

20. Number of Divisors
• If p is prime τ(p) = 2
• τ(2n) = 1,2, 22 ,… 2r,…. 2n
τ(2n)= n+1

21. Number of Divisors
n =
τ(n)=
In general
α1 + 1 α2 + 1 αr + 1
…..
0,1,2…α1 0,1,2…α2 0,1,2…αr
(α1 + 1)(α2 + 1) (αr + 1)…..

22. Number of Divisors
τ(n)= …..(α1 + 1)(α2 + 1) (αr + 1)
Example:
τ(180) = 22 * 32 * 5
τ(180) = (2+1)(2+1)(1+1)
τ(180) = (3)(3)(2)
τ(180) = 18
Do the prime factorization
Add one to each power
Then multiply the numbers

23. Number of Divisors
τ(n)= …..(α1 + 1)(α2 + 1) (αr + 1)
Example:
τ(360) = 23 * 32 * 5
τ(360) = (3+1)(2+1)(1+1)
τ(360) = (4)(3)(2)
τ(360) = 24
Example:
τ(540) = 22 * 33 * 5
τ(360) = (2+1)(3+1)(1+1)
τ(360) = (3)(4)(2)
τ(360) = 24

24. Sum of Divisors
σ(n) = sum of divisors function
σ(n) = sum of the divisors of n
Example:
σ(12)

25. Sum of Divisors
σ(n) = sum of divisors function
σ(n) = sum of the divisors of n
Example:
σ(12) = 1

26. Sum of Divisors
σ(n) = sum of divisors function
σ(n) = sum of the divisors of n
Example:
σ(12) = 1,2

27. Sum of Divisors
σ(n) = sum of divisors function
σ(n) = sum of the divisors of n
Example:
σ(12) = 1,2,3

28. Sum of Divisors
σ(n) = sum of divisors function
σ(n) = sum of the divisors of n
Example:
σ(12) = 1,2,3,4

29. Sum of Divisors
σ(n) = sum of divisors function
σ(n) = sum of the divisors of n
Example:
σ(12) = 1,2,3,4,6

30. Sum of Divisors
σ(n) = sum of divisors function
σ(n) = sum of the divisors of n
Example:
σ(12) = 1,2,3,4,6,12
σ(12) = 1+2+3+4+6+12
σ(12) = 28

31. Definition
The sum of divisors function is given by

32. Sum of Divisors
Example:
σ(180) =22 * 32 * 5
σ(180) =
σ(180) =
σ(180) =7 x 13 x 6
P1=2 σ1=2
P2=3 σ2=2
P3=5 σ3=1
σ(180) =546

33. Sum of Divisors
Example:
σ(360) =23 * 32 * 5
σ(360) =
σ(360) =
σ(360) =15 x 13 x 6
P1=2 σ1=3
P2=3 σ2=2
P3=5 σ3=1
σ(360) =1170

34. Proper Divisors
• The proper divisors of a number a are those
divisors that are less than the a
• Example:
15 -
14 -
20 -
1,3,5
1,2,7
1,2,4,5,10

35. - In symbol
- It is the sum of the proper divisors of a
“Sigma star of a”

36. Perfect Numbers
• A number is called perfect if it is equal to the
sum of its proper divisors.
• Example :
= 1+2+3
= 6

37. Other perfect numbers
• 6
• 28
• 496
• 8,128
• 33,550,336
• 8,589,869,056
• 137,438,691,328….

38. Deficient Numbers
• A number is called deficient if the sum of its
proper divisors is less than the number
Example:
= 1+2+4
= 7
7 < 8
Deficient Number

39. Other Deficient number
• 1,
• 2,
• 3,
• 4,
• 5,
• 7,
• 15,
• 16,
• 17,
• 19,
• 21,
• 22,
• 8,
• 9,
• 10,
• 11,
• 13,
• 14,
• 23,
• 25,
• 26,
• 27,
• 29,
• 31…

40. Abundant Numbers
• A number is called deficient if the sum of its
proper divisors is greater than the number
• Example:
= 1+2+3+4+6+8+12
= 36 36 >24
Abundant Number

41. Other abundant numbers
• 12,
• 18,
• 20,
• 24,
• 30,
• 36,
• 66,
• 70,
• 72,
• 78,
• 80,
• 84,
• 40,
• 42,
• 48,
• 54,
• 56,
• 60,
• 88,
• 90,
• 96,
• 100,
• 102,
• 104…

42. Amicable Numbers
• Two numbers a and b are amicable pair if
• Example:

43. Other Amicable numbers
• (1184, 1210),
• (2620, 2924)
• (5020, 5564),
• (6232, 6368),
• (10744, 10856),
• (12285, 14595),
• (17296, 18416),
• (63020, 76084)…

44. PARTITION FUNCTION

45. Partition Function
• A partition of a counting number N is an
expression that represents N as a sum of
(usually smaller) counting numbers.
• P(n), sometimes also denoted p(n) (Abramowitz and Stegun
1972, p. 825; Comtet 1974, p. 94; Hardy and Wright 1979, p. 273; Conway and Guy 1996, p. 94;
Andrews 1998, p. 1), gives the number of ways of writing
the integer as a sum of positive integers,
where the order of addends is not considered
significant. By convention, partitions are
usually ordered from largest to smallest (Skiena 1990,
p. 51).

46. PARTITION FUNCTION
• For example, there are eight partitions of the
number 4 if order is considered important:
• There are just five partitions of the number 4
if order is not considered important:
4 3+1 1+3 2+2
2+1+1 1+2+1 1+1+2 1+1+1+1
4 3+1 2+2 2+1+1 1+1+1+1

47. PARTITION FUNCTION
• One can place all sorts of restrictions on the
types of partitions one wishes to count.
• For example, there are eight partitions of the
number 10 with exactly three terms, order not
important:
4+ 3+ 3
8 +1+ 1 7 +2 +1 6+ 3+ 1 6 +2 +2
5 +4+ 1 5 +3+ 2 4+ 4+ 2

48. PARTITION FUNCTION
Example: how many partitions are there in 7
where no part is larger than 2?
P(m,n)
P(2,7)
P(2,7) = 4
2+2+2+1
2+2+1+1+1
2+1+1+1+1+1
1+1+1+1+1+1+1

49. PARTITION FUNCTION
Example: how many partitions are there in 7
whose least part is 2 ?
P(2,7)
P(2,7) = 4
7
2+5
3+4
2+3+3

50. Example: how many partitions are there in 7
where all parts are odd?
P(2,7)
P(2,7) = 4
7
5+1+1
3+1+1+1+1
1+1+1+1+1+1+1
PARTITION FUNCTION

51. Example: how many partitions are there in 7
where all parts are even?
P(2,7)
P(2,7) = 0
none
PARTITION FUNCTION

52. Partition Function
• Definition: a generating function for an
arithmetic function f(k) is the infinite
polynomial
•

53. Partition Function
• the generating function for p(n) is the product
of geometric series since is a geometric
series.
• Then,
= (1 + x1 + x2 + x3 + x4 + x5 + …)(1 + x2 +
x4 + x6 + x8 + …)(1 + x3 + x6 + x9+ …)(1 +
x4 + x8 + …)(1 + x5 + …)(1 + x6 + …)…

54. = (1 + x1 + x2 + x3 + x4 + x5 + …)(1 + x2 +
x4 + x6 + x8 + …)(1 + x3 + x6 + x9+ …)(1 +
x4 + x8 + …)(1 + x5 + …)(1 + x6 + …)…
Partition Function
= 1 + x1 + 2x2 + 3x3 + 5x4 + 7x5 + 11x6…

55. Partition Function
Number P(n) Number P(n)
0 1 11 56
1 1 12 77
2 2 13 101
3 3 14 135
4 5 15 176
5 7 16 231
6 11 17 297
7 15 18 385
8 22 19 490
9 30 20 627
10 42 21 792

56. Partition Function
• Generating function by Indian genius Srinivasa
Ramanujan.

57. Partition Function

58. END
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