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/extra/project-euler/069/069.factor

http://github.com/abeaumont/factor
Unknown | 84 lines | 60 code | 24 blank | 0 comment | 0 complexity | 4d57b2261e6032fead4b9b40c02e0a35 MD5 | raw file
 1! Copyright (c) 2009 Aaron Schaefer.
 2! See http://factorcode.org/license.txt for BSD license.
 3USING: combinators fry kernel math math.primes math.primes.factors math.ranges
 4    project-euler.common sequences ;
 5IN: project-euler.069
 6
 7! http://projecteuler.net/index.php?section=problems&id=69
 8
 9! DESCRIPTION
10! -----------
11
12! Euler's Totient function, φ(n) [sometimes called the phi function], is used
13! to determine the number of numbers less than n which are relatively prime to
14! n. For example, as 1, 2, 4, 5, 7, and 8, are all less than nine and
15! relatively prime to nine, φ(9)=6.
16
17!     +----+------------------+------+-----------+
18!     | n  | Relatively Prime | φ(n) | n / φ(n)  |
19!     +----+------------------+------+-----------+
20!     | 2  | 1                | 1    | 2         |
21!     | 3  | 1,2              | 2    | 1.5       |
22!     | 4  | 1,3              | 2    | 2         |
23!     | 5  | 1,2,3,4          | 4    | 1.25      |
24!     | 6  | 1,5              | 2    | 3         |
25!     | 7  | 1,2,3,4,5,6      | 6    | 1.1666... |
26!     | 8  | 1,3,5,7          | 4    | 2         |
27!     | 9  | 1,2,4,5,7,8      | 6    | 1.5       |
28!     | 10 | 1,3,7,9          | 4    | 2.5       |
29!     +----+------------------+------+-----------+
30
31! It can be seen that n = 6 produces a maximum n / φ(n) for n ≤ 10.
32
33! Find the value of n ≤ 1,000,000 for which n / φ(n) is a maximum.
34
35
36! SOLUTION
37! --------
38
39! Brute force
40
41<PRIVATE
42
43: totient-ratio ( n -- m )
44    dup totient / ;
45
46PRIVATE>
47
48: euler069 ( -- answer )
49    2 1000000 [a,b] [ totient-ratio ] map
50    [ supremum ] keep index 2 + ;
51
52! [ euler069 ] 10 ave-time
53! 25210 ms ave run time - 115.37 SD (10 trials)
54
55
56! ALTERNATE SOLUTIONS
57! -------------------
58
59! In order to obtain maximum n / φ(n), φ(n) needs to be low and n needs to be
60! high. Hence we need a number that has the most factors. A number with the
61! most unique factors would have fewer relatively prime.
62
63<PRIVATE
64
65: primorial ( n -- m )
66    {
67        { [ dup 0 = ] [ drop V{ 1 } ] }
68        { [ dup 1 = ] [ drop V{ 2 } ] }
69        [ nth-prime primes-upto ]
70    } cond product ;
71
72: primorial-upto ( limit -- m )
73    1 swap '[ dup primorial _ <= ] [ 1 + dup primorial ] produce
74    nip penultimate ;
75
76PRIVATE>
77
78: euler069a ( -- answer )
79    1000000 primorial-upto ;
80
81! [ euler069a ] 100 ave-time
82! 0 ms ave run time - 0.01 SD (100 trials)
83
84SOLUTION: euler069a