I’ve been doing the Weekly
Challenges. The
latest
involved prime-counting and fraction trees. (Note that this is open
until 9 January 2022.)

Task 1: 10001st Prime Number

Write a script to generate the 10001st prime number.

The fast way of generating small primes, the sieve of Eratosthenes,
requires an upper bound, so clearly what I need here is some sort of
prime-counting
function –
i.e. something to give me a number that is guaranteed to be no lower
than the 10001st prime. I thought about implementing the
Meissel-Lehmer algorithm but it seemed too much like work, so instead
I tweaked my existing sieve of Eratosthenes generator for better
performance, and used a relation from that Wikipedia page:

`P(sub n) < n log (n log n)`

where `n ≥ 6`

This isn't entirely true; it's proved up to n ≤ e^95, which is plenty
for this purpose. The result originates in Rosser, Barkley (1941).
"Explicit bounds for some functions of prime numbers". American
Journal of Mathematics. 63 (1): 211–232. doi:10.2307/2371291.
JSTOR 2371291. Which of course I can't legally read, because it's
*vitally important* that everyone who might want to look at an
eighty-year-old mathematical paper should first pay their two obols to
Johns Hopkins. Well, at least it's not Elsevier.

Anyway, I'll do that, then use the result as an upper bound for my
existing prime generator. Raku:

```
sub nthprime($n) {
my $m=15;
if ($n >= 6) {
$m=floor(1+$n*log($n*log($n)));
}
my @primes=genprimes($m);
return @primes[$n-1];
}
```

The prime generator is slightly tweaked from the version of two weeks
ago to use the `6n±1`

progression of prime candidates, which means a
short FIFO queue filled as necessary. (I suspect in Rust this could be
done with a custom iterator, but I just used `VecDeque`

.)

```
sub genprimes($mx) {
my @primes;
{
my $primesh=(2,3).SetHash;
loop (my $i=6;$i <= $mx+1; $i += 6) {
for ($i-1,$i+1) -> $j {
if ($j <= $mx) {
$primesh{$j}=True;
}
}
}
my $p=2;
my @q=[2,3,5,7];
my $mr=floor(sqrt($mx));
while ($p <= $mr) {
if ($primesh{$p}:exists) {
my $i=$p*$p;
while ($i <= $mx) {
$primesh{$i}:delete;
$i += $p;
}
}
if (@q.elems < 2) {
@q.push(@q[*-1]+4);
@q.push(@q[*-1]+2);
}
$p=@q.shift;
}
@primes=$primesh.keys.sort;
}
return @primes;
}
```

I have code for PostScript that ought to work, but ran off the end of
GhostScript's ability to manage memory.

Task 2: Curious Fraction Tree

Consider the following Curious Fraction Tree:

(image elided, see below)

You are given a fraction, member of the tree created similar to the above sample.

Write a script to find out the parent and grandparent of the given member.

This is a Calkin-Wilf
tree rather
than the Stern-Brocot that puts all its values in order. Every
language I'm using except for PostScript has something like object
orientation, so in each of them I defined a Fraction class, with
methods `set_from_string`

to set up the fraction from a string value,
`stringify`

to convert it *to* a string value, and `get_parent`

to
return a new Fraction object that's a parent of the current one. If I
were doing more with this, it would be easy to extend it:
`get_left_child`

, `get_right_child`

, `reduce`

, arithmetic operations,
and so on. Raku does actually offer a `Rat`

class, but it seemed
easier to write my own than to extend theirs.

In Lua:

```
Fraction = {n = 1, b = 1}
function Fraction:new (o)
o = o or {}
self.__index = self
setmetatable(o,self)
return o
end
function Fraction:get_parent ()
p=Fraction:new()
p.n = self.n
p.d = self.d
if p.n < p.d then
p.d = p.d - p.n
else
p.n = p.n - p.d
end
return p
end
function Fraction:stringify ()
return string.format("%d/%d",self.n,self.d)
end
function Fraction:set_from_string (s)
for n, d in string.gmatch(s,"(%d+)/(%d+)") do
self.n=tonumber(n)
self.d=tonumber(d)
end
return self
end
function p_gp(s)
f=Fraction:new():set_from_string(s)
out={}
for i = 1,2 do
f=f:get_parent()
table.insert(out,f:stringify())
end
return out
end
```

In PostScript I move the `set_from_string`

logic down inside `p_gp`

and just work with a pair of numbers on the stack. (I have library
functions to do things like pushing to an undefined-length array or
string (`apush`

and `spush`

), or to convert an integer to a string
without predefining the string's length (`i2s`

).

```
/stringify {
exch i2s 0 string exch spush
(/) spush
exch i2s spush
} bind def
/get_parent {
2 copy
gt {
exch 1 index sub exch
} {
1 index sub
} ifelse
} bind def
/p_gp {
/out 0 array def
(/) search {
cvi
exch pop
exch cvi
1 1 2 {
pop get_parent 2 copy stringify out exch apush /out exch def
} for
out
} {
pop out
} ifelse
} bind def
```

While it's not relevant here, the Fusc sequence that we saw in
challenge #104 is the sequence of numerators (or denominators) in a
breadth-first traversal of the tree.

And I'm adding JavaScript to the array of languages I'm doing this in.
(Running under Node.js for command-line testing, but I'm trying to
avoid anything that wouldn't run as easily in a browser.) Where will
this madness end? They each have a reason for me to learn to use them,
though, and solving these problems in multiple languages has
definitely helped get me loose from the regexes-and-hashes approach
that doing everything in Perl encourages.

Full code on
github.

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