85 lines
2.3 KiB
Factor
85 lines
2.3 KiB
Factor
! Copyright (c) 2009 Aaron Schaefer.
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! See http://factorcode.org/license.txt for BSD license.
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USING: combinators fry kernel math math.primes math.primes.factors math.ranges
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project-euler.common sequences ;
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IN: project-euler.069
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! http://projecteuler.net/index.php?section=problems&id=69
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! DESCRIPTION
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! -----------
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! Euler's Totient function, φ(n) [sometimes called the phi function], is used
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! to determine the number of numbers less than n which are relatively prime to
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! n. For example, as 1, 2, 4, 5, 7, and 8, are all less than nine and
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! relatively prime to nine, φ(9)=6.
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! +----+------------------+------+-----------+
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! | n | Relatively Prime | φ(n) | n / φ(n) |
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! +----+------------------+------+-----------+
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! | 2 | 1 | 1 | 2 |
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! | 3 | 1,2 | 2 | 1.5 |
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! | 4 | 1,3 | 2 | 2 |
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! | 5 | 1,2,3,4 | 4 | 1.25 |
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! | 6 | 1,5 | 2 | 3 |
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! | 7 | 1,2,3,4,5,6 | 6 | 1.1666... |
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! | 8 | 1,3,5,7 | 4 | 2 |
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! | 9 | 1,2,4,5,7,8 | 6 | 1.5 |
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! | 10 | 1,3,7,9 | 4 | 2.5 |
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! +----+------------------+------+-----------+
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! It can be seen that n = 6 produces a maximum n / φ(n) for n ≤ 10.
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! Find the value of n ≤ 1,000,000 for which n / φ(n) is a maximum.
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! SOLUTION
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! --------
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! Brute force
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<PRIVATE
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: totient-ratio ( n -- m )
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dup totient / ;
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PRIVATE>
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: euler069 ( -- answer )
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2 1000000 [a,b] [ totient-ratio ] map
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[ supremum ] keep index 2 + ;
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! [ euler069 ] 10 ave-time
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! 25210 ms ave run time - 115.37 SD (10 trials)
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! ALTERNATE SOLUTIONS
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! -------------------
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! In order to obtain maximum n / φ(n), φ(n) needs to be low and n needs to be
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! high. Hence we need a number that has the most factors. A number with the
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! most unique factors would have fewer relatively prime.
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<PRIVATE
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: primorial ( n -- m )
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{
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{ [ dup 0 = ] [ drop V{ 1 } ] }
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{ [ dup 1 = ] [ drop V{ 2 } ] }
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[ nth-prime primes-upto ]
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} cond product ;
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: primorial-upto ( limit -- m )
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1 swap '[ dup primorial _ <= ] [ 1 + dup primorial ] produce
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nip penultimate ;
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PRIVATE>
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: euler069a ( -- answer )
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1000000 primorial-upto ;
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! [ euler069a ] 100 ave-time
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! 0 ms ave run time - 0.01 SD (100 trials)
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SOLUTION: euler069a
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