# RogerBW's Blog

 The Weekly Challenge 159: Farey and Moebius 07 April 2022 I’ve been doing the Weekly Challenges. The latest involved generating more sequences. (Note that this is open until 10 April 2022.) Task 1: Farey Sequence You are given a positive number, `\$n`. Write a script to compute Farey Sequence of the order `\$n`. In other words, all the fractions from 0 to 1 with integral numerators and denominators up to `\$n`, in increasing order of value, in reduced forms (so you list 1/2 but not 2/4). Of course some people might use floating point numbers for this. But floating point is bad and wrong. So instead I work out the LCM for the series of denominators (2 up to n), using the code from #155, then store each fraction as a pair of numbers, keyed by (value × lcm), which will always be an integer. In JavaScript: ``````function farey (n) { `````` Calculate the lcm, just once. `````` let l=lcmseries(2,n) `````` Initialise the hash that'll hold the fractions, and the array that'll hold the keys for later sorting. `````` let d=new Map() let s=[] `````` Iterate denominators from 1 to n. `````` for (let i=1;i <= n;i++) { `````` Calculate the multiplier value for this denominator. `````` let m=l/i `````` Iterate numerators from 0 to denominator. `````` for (let j=0;j <= i;j++) { `````` The value of this fraction, multiplied by the lcm. `````` let k=m*j `````` If we haven't already seen this value, stick it in the hash, as a two-element array. (This takes care of the reduced forms; if n=4, 2/4 produces a k value of 2, but we already got a 2 for 1/2.) Also push the key value onto its list. `````` if (!d.has(k)) { d.set(k,[j,i]) s.push(k) } } } `````` Sort the key list. (Some languages, like this one, prefer to sort in place; for others I don't need a separate key list, and I just sort the keys of the hash and output in that order.) `````` s.sort(function(a,b) { return a-b; }) `````` Then return the hash values in order of the sorted key list. `````` return s.map(i => d.get(i)) } `````` Kotlin has a `Pair` class, and Raku has `Rat` for storing fractions without the hazards of floating point, but otherwise I just use two-element arrays. Task 2: Moebius Number You are given a positive number `\$n`. Write a script to generate the Moebius Number for the given number. Please refer to wikipedia page for more informations. Which comes down to: looking through the list of prime factors, if there are any squares or higher powers, the result is 0; otherwise it's +1 if the number of prime factors is even, -1 if it's odd. Now obviously I could take a short cut by exiting early from the prime factorisation algorithm, but `primefactors` is useful library code, so I'll lose a little efficiency by calculating them in full and then analysing the result. (I did make the improvement to this algorithm I mentioned back in 155.) In Perl (reusing the standard `genprimes` of course): ``````sub primefactor { my \$n=shift; my %f; my \$m=\$n; foreach my \$p (@{genprimes(int(sqrt(\$n)))}) { while (\$m % \$p == 0) { \$f{\$p}++; \$m=int(\$m/\$p); if (\$m == 1) { last; } } } if (\$m > 1) { \$f{\$m}++; } return \%f; } `````` Then the actual Möbius analysis is nearly trivial: ``````sub moebius { my \$n=shift; my \$z=0; foreach my \$v (values %{primefactor(\$n)}) { if (\$v>1) { return 0; } \$z++; } if (\$z % 2 == 0) { return 1; } return -1; } `````` Full code on github. Comments on this post are now closed. If you have particular grounds for adding a late comment, comment on a more recent post quoting the URL of this one. 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