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

85 lines
2.3 KiB
Factor

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