Initial data commit

Task/M-bius-function/Jq/m-bius-function-3.jq

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+# relatively_prime(previous) tests whether the input integer is prime
+# relative to the primes in the array "previous":
+def relatively_prime(previous):
+  . as $in
+  | (previous|length) as $plen
+  # state: [found, ix]
+  | [false, 0]
+  | until( .[0] or .[1] >= $plen;
+           [ ($in % previous[.[1]]) == 0, .[1] + 1] )
+  | .[0] | not ;
+
+# Emit a stream in increasing order of all primes (from 2 onwards)
+# that are less than or equal to mx:
+def primes(mx):
+  # The helper function, next, has arity 0 for tail recursion optimization;
+  # it expects its input to be the array of previously found primes:
+  def next:
+     . as $previous
+     | ($previous | .[length-1]) as $last
+     | if ($last >= mx) then empty
+       else ((2 + $last)
+       | until( relatively_prime($previous) ; . + 2)) as $nextp
+       | if $nextp <= mx
+         then $nextp, (( $previous + [$nextp] ) | next)
+	 else empty
+         end
+       end;
+  if mx <= 1 then empty
+  elif mx == 2 then 2
+  else (2, 3, ([2,3] | next))
+  end ;
+
+# Return an array of the distinct prime factors of . in increasing order
+def prime_factors:
+
+  # Return an array of prime factors of . given that "primes"
+  # is an array of relevant primes:
+  def pf($primes):
+    if . <= 1 then []
+    else . as $in
+    | if ($in | relatively_prime($primes)) then [$in]
+      else reduce $primes[] as $p
+             ([];
+              if ($in % $p) != 0 then .
+ 	      else . + [$p] +  (($in / $p) | pf($primes))
+	      end)
+      end
+      | unique
+    end;
+
+  if . <= 1 then []
+  else . as $in
+  | pf( [ primes( (1+$in) | sqrt | floor)  ] )
+  end;