RogerBW's Blog

The Weekly Challenge 173: The Aesthetics of Sylvester 14 July 2022

I’ve been doing the Weekly Challenges. The latest involved more mathematical tests. (Note that this is open until 17 July 2022.)

Task 1: Esthetic Number

You are given a positive integer, $n.

Write a script to find out if the given number is Esthetic Number.

I.e. each digit differs from its predecessor by 1.

When this is touched on elsewhere, the number base seems to be important, so I wrote a base-10 wrapper. Raku:

sub esthetic10($n) {
    return esthetic($n,10);

sub esthetic($n0,$base) {
    my $n = $n0;

Set up previous digit and check flag.

    my $pdigit;
    my $ch = False;
    while ($n > 0) {

Extract the rightmost remaining digit of the number.

        my $digit = $n % $base;

If the check flag is set (which it won't be for the first time through the loop), compare it with the previous digit, and return false if they aren't 1 apart.

        if ($ch && abs($digit - $pdigit) != 1) {
            return False;

Set the check flag, store the digit as previous digit, and divide $n for the next pass.

        $ch = True;
        $pdigit = $digit;
        $n = floor($n / $base);

If all the tests passed, the number as a whole passes.

    return True;

The other languages are more or less the same: some of them relegate abs() to a maths library.

Task 2: Sylvester's sequence

Write a script to generate first 10 members of Sylvester's sequence.

which starts with 2, and x(n+1) is the product of x(0) to x(n), plus 1.

Which offers an optimisation, because x(n) is itself the product of all previous values plus 1, so we only need one multiplication per term:

x(n+1) = x(n) * (x(n) - 1) + 1

Since the tenth value is something like 2^346, this needs some sort of arbitrary-sized integer support. PostScript and Lua don't offer it at all (there are various third-party libraries for Lua), so I'm not writing code for them. Ruby, Python and Raku transparently use large integers as needed, and Perl can be told to. JavaScript has recently (ES2020) gained the BigInt type which just needs an "n" at the end of the literal, and can't be readily mixed with standard integers. Kotlin has BigInteger, which works similarly. And Rust has several options, but num_bigint seems to be the most widely used.

Once that's been resolved, the rest is easy. Ruby:

def sylvester(ct)
  o = [ 2 ]
  2.upto(ct) do
    o.push(1 + (o[-1] * (o[-1] - 1)))
  return o

Full code on github.

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