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# Untitled

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1. def divides(n,d):
2.     '''True if remainder of n/d is 0, False otherwise'''
3.     return n%d == 0
4.
5. def least_divisor_from(n,k):
6.     '''smallest number starting from k, k <= n, that divides n'''
7.     if divides(n,k):return k
8.     else: return least_divisor_from(n,k+1)
9.
10. def least_divisor(n):
11.     '''smallest divisor of n that is greater than 1'''
12.     return least_divisor_from(n,2)
13.
14. def is_prime(n):
15.     '''True if n is prime, False otherwise'''
16.     return least_divisor(n)== n
17.
18. def next_prime(n):
19.     '''next prime > n'''
20.     if is_prime(n+1): return n+1
21.     else: return next_prime(n+1)
22.
23.
24. def primes_between(a,b):
25.     '''list of primes in [a..b]'''
26.     if a>b: return []
27.     if is_prime(a): return [a]+primes_between(next_prime(a),b)
28.     else:  return primes_between(next_prime(a),b)
29.
30. def prime_factors(n):
31.     '''list of prime factors of n'''
32.
33. def factorial(n):
34.     '''returns n*(n-1)*(n-2)*...*2*1'''
35.     if n==0:return 1
36.     else:return n* factorial(n-1)
37.
38. def P(n,r):
39.     '''returns nPr'''
40.     if r == 0: return 1
41.     else: return n*P(n-1,r-1)
42.
43. def C(n,r):
44.     '''returns nCr'''
45.     return P(n,r)//factorial(r)
46.
47. def gcf(a,b):
48.     '''greatest common factor of a and b'''
49.     if a==b: return a
50.     if a<b: return gcf(a,b-a)
51.     if a>b: return gcf(a-b,b)
52.
53. def lcm(a,b):
54.     '''least common multiple of a and b'''
55.     return a*b//gcf(a,b
56.
57. def arithmetic(a,d,n):
58.     '''arithmetic sequence of n terms'''
59.     if n == 0: return []
60.     else: return [a]+arithmetic(a+d,d,n-1)
61.
62. def geometric(g,r,n):
63.     '''geometric sequence of n terms'''
64.     if n==0: return []
65.     else: return [g]+geometric(g*r,r,n-1)
66.
67. def first(L):
68.     '''first item in list L'''
69.     return L[0]
70.
71. def rest(L):
72.     '''all items in list L after the first'''
73.     return L[1:]
74.
75. def last(L):
76.     '''last item in list L'''
77.     return rest(L)
78.
79. def init(L):
80.     '''all items in lst before the last'''
81.
82. def length(L):
83.     '''number of items in list L'''
84.     if L == []: return 0
85.     else: return 1+length(rest(L))
86.
87. def sum(L):
88.     '''sum of items in list L'''
89.
90. def product(L):
91.     '''product of items in list L'''
92.
93. def mean(L):
94.     '''arithmetic average of list L'''
95.
96. def smaller(a,b):
97.     '''smaller of values a and b'''
98.
99. def min(L):
100.     '''minimum value in lst'''
101.
102. def larger(a,b):
103.     '''larger of values a and b'''
104.
105. def max(L):
106.     '''maximum value in list L'''
107.
108. def remove(item, L):
109.     '''list L without item'''
110.
111. def sorted(L):
112.     '''items of list L sorted in increasing order'''
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