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# The Weekly Challenge 173

You can find more information about this weeks, and previous weeks challenges at:

  https://theweeklychallenge.org/

If you are not already doing the challenge - it is a good place to practise your
**perl** or **raku**. If it is not **perl** or **raku** you develop in - you can
submit solutions in whichever language you feel comfortable with.

You can find the solutions here on github at:

https://github.com/drbaggy/perlweeklychallenge-club/tree/master/challenge-173/james-smith

# Task 1 - Esthetic Number

***You are given a positive integer, `$n`. Write a script to find out if the given number is Esthetic Number. An esthetic number is a positive integer where every adjacent digit differs from its neighbour by 1.***

## Solution

We split the number up into it's individual digits. We then compare each digit with the previous one, if they are `+1` or `-1` we then update the "last" digit and repeat otherwise we short cut the loop and return 0.

```perl
sub is_esthetic {
  my($f,@n) = split //, pop;
  abs( $_-$f ) == 1 ? ($f=$_) : return 0 for @n;
  1
}
```

# Task 2 - Sylvester’s sequence

***Write a script to generate first 10 members of Sylvester's sequence. For each member of the sequence the value is equal to the product of previous terms + 1***

## Solution

We note that `S(n)` is the product of all prior terms plus 1, and so `S(n-1)` is similar. So this gives us that `S(n+1) = S(n)*( S(n) - 1 ) + 1`.

This reduces our problem considerably and we have the following code. To get the values our for `n` bigger than 7 we have to resort the the `bigint` directive.

```perl
my $n=2;
say for 2, map{ $n*=$n-1; ++$n } 1..9;
```

## Answers
```
 1                                                                                                                                            2
 2                                                                                                                                            3
 3                                                                                                                                            7
 4                                                                                                                                           43
 5                                                                                                                                        1,807
 6                                                                                                                                    3,263,443
 7                                                                                                                           10,650,056,950,807
 8                                                                                                          113,423,713,055,421,844,361,000,443
 9                                                                       12,864,938,683,278,671,740,537,145,998,360,961,546,653,259,485,195,807
10  165,506,647,324,519,964,198,468,195,444,439,180,017,513,152,706,377,497,841,851,388,766,535,868,639,572,406,808,911,988,131,737,645,185,443
```

## Calculation showing S(n+1) can be written in terms of S(n) only
```
               S(n) = S(n-1)S(n-2)...S(1) + 1'
S(n-1)S(n-2)...S(1) = S(n) - 1

             S(n+1) = S(n)S(n-1)S(n-2)...S(1) + 1'
             S(n+1) = S(n)( S(n) - 1 ) + 1
```